Global Analysis Agricola&Friedrich

Global AnalysisDifferential Forms in Analysis,Geometry and Physics

Ilka AgricolaThomas Friedrich

Graduate Studiesin MathematicsVolume 52

American Mathematical Society

Global Analysis

Global AnalysisDifferential Forms in Analysis,Geometry and Physics


Translated byAndreas Nestke

Graduate Studiesin Mathematics

Volume 52

American Mathematical SocietyProvidence, Rhode Island

Editorial BoardWalter CraigNikolai Ivanov

Steven G. KrantzDavid Saltman (Chair)

2000 Mathematics Subject Classification. Primary 53-01;Secondary 57-01, 58-01, 22-01, 74-01, 78-01, 80-01, 35-01.

This book was originally published in German by Friedr. Vieweg & Sohn Verlagsge-sellschaft mbH, D-65189 Wiesbaden, Germany, as "Ilka Agricola and Thomas Friedrich:Globale Analysis. 1. Auflage (1st edition)", ©F}iedr. Vieweg & Sohn VerlagsgesellschaftmbH, Braunschweig/Wiesbaden, 2001

Translated from the German by Andreas Nestke

Library of Congress Cataloging-in-Publication DataAgricola, I1ka, 1973-

(Globale Analysis. English)Global analysis : differential forms in analysis, geometry, and physics / Ilka Agricola, Thomas

Ftiedrich ; translated by Andreas Nestke.p. cm. - (Graduate studies in mathematics, ISSN 1065-7339 ; v. 52)

Includes bibliographical references and index.ISBN 0-8218-2951-3 (alk. paper)1. Differential forms. 2.Mathematical physics. 1. Friedrich, Thomas, 1949 ll. Title.

111. Series.

QA381.A4713 2002514'.74- dc2l 2002027681

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence. Rhode Island 02904-2294, USA. Requests can also be made bye-mail to reprint-peraissionlaes.org.

© 2002 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rights

except those granted to the United States Government.Printed in the United States of America.

The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://wv.aas.org/

10987654321 070605040302

Preface

This book is intended to introduce the reader into the world of differentialforms and, at the same time, to cover those topics from analysis, differ-ential geometry, and mathematical physics to which forms are particularlyrelevant. It is based on several graduate courses on analysis and differen-tial geometry given by the second author at Humboldt University in Berlinsince the beginning of the eighties. From 1998 to 2000 the authors taughtboth courses jointly for students of mathematics and physics, and seized theopportunity to work out a self-contained exposition of the foundations ofdifferential forms and their applications. In the classes accompanying thecourse, special emphasis was put on the exercices, a selection of which thereader will find at the end of each chapter. Approximately the first half ofthe book covers material which would be compulsary for any mathematicsstudent finishing the first part of his/her university education in Germany.The book can either accompany a course or be used in the preparation ofseminars.

We only suppose as much knowledge of mathematics as the reader would ac-quire in one year studying mathematics or any other natural science. Fromlinear algebra, basic facts on multilinear forms are needed, which we brieflyrecall in the first chapter. The reader is supposed to have a more extensiveknowledge of calculus. Here, the reader should be familiar with differentialcalculus for functions of several variables in euclidean space R", the Rie-mann integral and, in particular, the transformation rule for the integral,as well as the existence and uniqueness theorem for solutions of ordinarydifferential equations. It is a reader with these prerequisites that we have inmind and whom we will accompany into the world of vector analysis, Pfaf flan

v

vi Preface

systems, the differential geometry of curves and surfaces in euclidean space,Lie groups and homogeneous spaces, symplectic geometry and mechanics,statistical mechanics and thermodynamics and, eventually, electrodynamics.

In Chapter 2 we develop the differential and integral calculus for differentialforms defined on open sets in euclidean space. The central result is Stokes'formula turning the integral of the exterior derivative of a differential formover a singular chain into an integral of the form itself over the bound-ary of the chain. This is in fact a far-reaching generalization of the maintheorem of differential and integral calculus: differentiation and integrationare mutually inverse operations. At the end of a long historical develop-ment mathematicians reached the insight that a series of important integralformulas in vector analysis can be obtained by specialization from Stokes'formula. We will show this in the second chapter and derive in this wayGreen's first and second formula, Stokes' classical formula, and Cauchy'sintegral formula for complex differentiable functions. Furthermore, we willdeduce Brouwer's fixed point theorem from Stokes' formula and the Weier-strass approximation theorem.

In Chapter 3 we restrict the possible integration domains to "smooth"chains. On these objects, called manifolds, a differential calculus; for func-tions and forms can be developed. Though we only treat submanifolds ofeuclidean space, this section is formulated in a way to hold for every Rie-mannian manifold. We discuss the concept of orientation of a ma.nifold, itsvolume form, the divergence of a vector field as well as the gradient and theLaplacian for functions. We then deduce from Stokes' formula the remainingclassical integral formulas of Riemannian geometry (Gauss-Ostrogradski for-mula, Green's first and second formula) as well as the Hairy Sphere theorem,for which we decided to stick to its more vivid German name, `'Hedgehogtheorem". A separate section on the Lie derivative of a differential formleads us to the interpretation of the divergence of a vector field as a mea-sure for the volume distortion of its flow. We use the integral formulas tosolve the Dirichlet problem for the Laplace equation on the ball in euclideanspace and to study the properties of harmonic functions on R". For thesewe prove, among other things, the maximum principle and Liouville's the-orem. Finally we discuss the Laplacian acting on forms over a Riemannianmanifold, as well as the Hodge decomposition of a differential form. Thisis a generalization of the splitting of a vector field with compact support inR" into the sum of a gradient field and a divergence-free vector field, goingback to Helmholtz. In the final chapter we prove Helmholtz' theorem withinthe framework of electrodynamics.

Preface vii

Apart from Stokes' theorem, the integrability criterion of Frobenius is one ofthe fundamental results in the theory of differential forms. A geometric dis-tribution (Pfafflan system) is defined by choosing a k-dimensional subspacein each tangent space of an n-dimensional manifold in a smooth way. Ageometric distribution can alternatively be described as the zero set of a setof linearly independent 1-forms. What one is looking for then is an answerto the question of whether there exists a k-dimensional submanifold suchthat, at each point, the tangent space coincides with the value of the givengeometric distribution. Frobenius' theorem gives a complete solution to thisquestion and provides a basic tool for the integration of certain systems offirst order partial differential equations. Therefore, Chapter 4 is devoted toa self-contained and purely analytical proof of this key result, which, more-over, will be needed in the sections on surfaces, symplectic geometry, andcompletely integrable systems.

Chapter 5 treats the differential geometry of curves and surfaces in euclideanspace. We discuss the curvature and the torsion of a curve, Frenet's formu-las, and prove the fundamental theorem of the theory of curves. We thenturn to some special types of curves and conclude this section by a proof ofFenchel's inequality. This states that the total curvature of a closed spacecurve is at least 27r. Surface theory is treated in Cartan's language of mov-ing frames. First we describe the structural equations of a surface, and thenwe prove the fundamental theorem of surface theory by applying Frobenius'theorem. The latter is formulated with respect to a frame adapted to thesurface and the resulting 1-forms. Next we start the tensorial description ofsurfaces. The first and second fundamental forms of a surface as well as therelations between them as expressed in the Gauss and the Codazzi-Mainardiequations are the central concepts here. We reformulate the fundamentaltheorem in this tensorial description of surface theory. Numerous examples(surfaces of revolution, general graphs and, in particular, reliefs, i.e. thegraph of the modulus of an analytic function, as well as the graphs of theirreal and imaginary parts) illustrate the differential-geometric treatment ofsurfaces in euclidean space. We study the normal map of a surface and arethus lead to its Gaussian curvature, which by Gauss' Theorema Egregiumbelongs to the inner geometry. Using Stokes' theorem, we prove the Gauss-Bonnet formula and an analogous integral formula for the mean curvatureof a compact oriented surface, going back to Steiner and Minkowski. An im-portant class of surfaces are minimal surfaces. Their normal map is alwaysconformal, and this observation leads to the so-called Weierstrass formu-las. These describe the minimal surface locally by a pair of holomorphicfunctions. Then we turn to the study of geodesic curves on surfaces, the in-tegration of the geodesic flow using first integrals as well as the investigation

of maps between surfaces. Chapter 5 closes with an outlook on the geometryof pseudo-Riemannian manifolds of higher dimension. In particular, we lookat Einstein spaces, as well as spaces of constant curvature.

Symmetries play a fundamental role in geometry and physics. Chapter 6contains an introduction into the theory of Lie groups and homogeneousspaces. We discuss the basic properties of a Lie group, its Lie algebra, andthe exponential map. Then we concentrate on proving the fact that everyclosed subgroup of a Lie group is a Lie group itself, and define the structureof a manifold on the quotient space. Many known manifolds arise as homo-geneous spaces in this way. With regard to later applications in mechanics,we study the adjoint representation of a Lie group.

Apart from Riemannian geometry, symplectic geometry is one of the es-sential pillars of differential geometry, and it is particularly relevant to theHamiltonian formulation of mechanics. Examples of symplectic manifoldsarise as cotangent bundles of arbitrary manifolds or as orbits of the coadjointrepresentation of a Lie group. We study this topic in Chapter 7. First weprove the Darboux theorem stating that all symplectic manifolds are locallyequivalent. Then we turn to Noether's theorem and interpret it in terms ofthe moment map for Hamiltonian actions of Lie groups on symplectic man-ifolds. Completely integrable Hamiltonian systems are carefully discussed.Using Frobenius' theorem, we demonstrate an algorithm for finding the ac-tion and angle coordinates directly from the first integrals of the Hamiltonfunction. In §7.5, we sketch the formulations of mechanics according toNewton, Lagrange, and Hamilton. In particular, we once again return toNoether's theorem within the framework of Lagrangian mechanics, whichwill be applied, among others, to integrate the geodesic flow of a pseudo-Riemannian manifold. Among the exercises of Chapter 7, the reader willfind some of the best known mechanical systems.

In statistical mechanics, particles are described by their position probabilityin space. Therefore one is interested in the evolution of statistical states ofa Hamiltonian system. In Chapter 8 we introduce the energy and informa-tion entropy for statistical equilibrium states. Then we characterize Gibbsstates as those of maximal information entropy for fixed energy, and provethat the microcanonical ensemble realizes the maximum entropy among allstates with fixed support. By means of the Gibbs states, we assign a ther-modynamical system in equilibrium to a Hamiltonian system with auxiliaryparameters satisfying the postulates of thermodynamics. We discuss the

Preface ix

role of pressure and free energy. A series of examples, like the ideal gas,solid bodies, and cycles, conclude Chapter 8.

Chapter 9 is devoted to electrodynamics. Starting from the Maxwell equa-tions, formulated both for the electromagnetic field strengths and for thedual 1-forms, we first deal with the static electromagnetic field. We provethe solution formula. for the inhomogeneous Laplace equation in three-spaceand obtain, apart from a description of the electric and the magnetic fieldin the static case, at the same time a proof for Helmholtz' theorem as men-tioned before. Next we turn to the vacuum electromagnetic field. Here weprove the solution formula for the Cauchy problem of the wave equation indimensions two and three. The chapter ends with a relativistic formulationof the Maxwell equations in Minkowski space, a discussion of the Lorentzgroup. the Maxwell stress tensor and a thorough treatment of the Lorentzforce.

We are grateful to Ms. Heike Pahlisch for her extensive work on the prepara-tion of the manuscript and the illustrations of the German edition. We alsothank the students in our courses from 1998 to 2000 for numerous commentsleading to additions and improvements in the manuscript. In particular,Dipl.-Math. Uli Kriihmer pointed out corrections in many chapters. Notleast our thanks are due to M. A. Claudia Frank for her thourough readingand correcting of the German manuscript with regard to language.

The English version at hand does not differ by much from the original Ger-man edition. Besides small corrections and additions, we included a detaileddiscussion of the Lorentz force and related topics in Chapter 9. Finally, wethank Dr. Andreas Nestke for his careful translation.

Berlin, November 2000 and May 2002


Contents

Preface v

Chapter 1. Elements of Multilinear Algebra

Exercises

1

8

Chapter 2. Differential Forms in R" 11

§2.1. Vector Fields and Differential Forms 11

§2.2. Closed and Exact Differential Forms 18

§2.3. Gradient, Divergence and Curl 23

§2.4. Singular Cubes and Chains 26

§2.5. Integration of Differential Forms and Stokes' Theorem 30

§2.6. The Classical Formulas of Green and Stokes 35

§2.7. Complex Differential Forms and Holomorphic Functions 36

§2.8. Brouwer's Fixed Point Theorem 38

Exercises 43

Chapter 3. Vector Analysis on Manifolds 47

§3.1. Submanifolds of lR' 47

§3.2. Differential Calculus on Manifolds 54

§3.3. Differential Forms on Manifolds 67

§3.4. Orientable Manifolds 69

§3.5. Integration of Differential Forms over Manifolds 76

§3.6. Stokes' Theorem for Manifolds 79

§3.7. The Hedgehog Theorem (Hairy Sphere Theorem) 81

xi

xii Contents

§3.8. The Classical Integral Formulas 82

§3.9. The Lie Derivative and the Interpretation of the Divergence 87

§3.10. Harmonic Functions 94

§3.11. The Laplacian on Differential Forms 100

Exercises 105

Chapter 4. Pfaffian Systems 111

§4.1. Geometric Distributions 111

§4.2. The Proof of Frobenius' Theorem 116

§4.3. Some Applications of Frobenius' Theorem 120

Exercises 126

Chapter 5. Curves and Surfaces in Euclidean 3-Space 129

§5.1. Curves in Euclidean 3-Space 129

§5.2. The Structural Equations of a Surface 141

§5.3. The First and Second Fundamental Forms of a Surface 147

§5.4. Gaussian and Mean Curvature 155

§5.5. Curves on Surfaces and Geodesic Lines 172

§5.6. Maps between Surfaces 180

§5.7. Higher-Dimensional Riemannian Manifolds 183

Exercises 198

Chapter 6. Lie Groups and Homogeneous Spaces 207

§6.1. Lie Groups and Lie Algebras 207

§6.2. Closed Subgroups and Homogeneous Spaces 215

§6.3. The Adjoint Representation 221

Exercises 226

Chapter 7. Symplectic Geometry and Mechanics 229

§7.1. Symplectic Manifolds 229

§7.2. The Darboux Theorem 236

§7.3. First Integrals and the Moment Map 238

§7.4. Completely Integrable Hamiltonian Systems 241

§7.5. Formulations of Mechanics 252

Exercises 264

Chapter 8. Elements of Statistical Mechanics and Thermodynam: .es 271

§8.1. Statistical States of a Hamiltonian System 271

Contents xiii

§8.2. Thermodynamical Systems in Equilibrium 283

Exercises 292

Chapter 9. Elements of Electrodynamics 295

§9.1. The Maxwell Equations 295

§9.2. The Static Electromagnetic Field 299

§9.3. Electromagnetic Waves 304

§9.4. The Relativistic Formulation of the Maxwell Equations 311

§9.5. The Lorentz Force 317

Exercises 325

Bibliography 333

Symbols 337

Index 339

Chapter 1

Elements ofMultilinear Algebra

Consider an n-dimensional vector space V over the field K of real or complexnumbers. Its dual space V' consists of all linear maps from V to K. Moregenerally, a multilinear and antisymmetric map,

wk: Vx...xV-+K.depending on k vectors from the vector space V, is called an exterior (mul-tilinear) form of degree k. The antisymmetry of wk means that, for allk vectors vl, ... , vk from V and any permutation a E Sk of the numbers{ 1, ... , k}, the following equation holds:

wk(VQ(1), ....Va(k)) = sgn(a) i.Jk(VI, -..,Vk).

Here sgn(a) denotes the sign of the permutation a. In particular, wk changessign under a transposition of the indices i and j:

wk(i1i ...,Vi, ...,Vj, ...,Vk) = -wk(v1, ...,v ....,VI.....Vk).

The vector space of all exterior k-forms will be denoted by Ak( V*). Fur-thermore, we will use the conventions A°(V') = K and A1(V*) = V.

Fixing an arbitrary basis e1, ... , e in the n-dimensional vector space V, wesee that each exterior k-form wk is uniquely determined by its values on allk-tuples of the form el,,... 441 where the indices are always supposed tobe strictly ordered, I = (i1 < ... < ik). On the other hand, a k-form can bedefined by arbitrarily prescribing its values on all ordered k-tuples of basisvectors and extending it to all k-tuples of vectors in an antisymmetric and

I

2 1. Elements of 1llultilinear Algebra

multilinear way. The number of different k-tuples of n elements is equal to(k) = k. Thus we conclude

Theorem 1. If k > n, Ak(V*) consists only of the zero map. For k < nthe dimension of the vector space Ak(V*) is equal to

dim (nk(V`)) _ (k)

Exterior forms can by multiplied, and the product is again an exterior form.

Definition 1. Let wk E Ak(V*) and 7/ E A1(V') be two exterior forms ofdegrees k and 1. respectively. Then the exterior product wk n 171 is defined asa (k + 1)-form by the formula

1wknll!(t'1....L'k+l) = Esgn(v)wk(v,(1),...Vo(k))11l(z'o(k+l)....vn(k+l))

k!l!oESk+j

Obviously. wk n,1 is a multilinear and antisymmetric map acting on (k + 1)vectors, i. e. of degeree k+l. The following theorem summarizes the algebraicrules governing computations involving the exterior multiplication of forms.

Theorem 2. The exterior product has the following properties:

(1) (wi +w2)AY/=w1 nrft+w2n

(2)

(3) (awk)A1)l=wkA(ar/)=a(wkAr/)foranyaEIfs:(4) (wkAgl)A m=wkA(r11A ');

(5) wk A 111 = (-1)k'711 n wk.

Proof. Only the last two formulas require a proof, in which we shall omitsome of the upper indices for better readability. First

(k+l)!m! (wk A11l) Aµ/'"(v1, ...,vk+l+m)1: sgn(a)(wk A 711)(v,(1), ... , v,(k+1))1f (vn(k+1+1).... , vs(k+l+m))

f7E Sk+l+m

Decompose the permutation group Sk+l+m into residue classes with respectto the subgroup Sk+l C Sk+1+m formed by all permutations acting as theidentity on the last m indices {k + 1 + 1, ..., k + l + m}. Each residueclass R thus consists of all permutations a E Sk+l+m with fixed valuesQ(k + I + 1), ..., v(k + I + m). Fixing any permutation ao E R. all theremaining elements a E R are parametrized by the elements in Sk+l:

or = a0 0 7r, 7r E Sk+l

1. Elements of Multilinear Algebra 3

Hence,

E sgn(o)(wk A rll)(Va(l), - - - , Vo(k+l))11m(Va(k+1+l), - - - , Va(k+l+m))oER

rsgn(oo)sgn(ir)(w A r/)(Vaoo,r(1), --Vooo,,(k+())1L(Vao(k+t+1), ..Vao(k+t+m)),rESk+i

= sgn(ao)(k + l)!(w A 17)(vao(1), ... , Voo(k+1))1J(VCo(k+1+1), ... , Vao(k+1+m))-

Using now the definition of the exterior product wk A 711, this leads to theformula

k! 1!

E(k + l)! aRsgn(a)(w A i))(va(1), vo(k+l+m))

= sgn(o)w(va(1), ... va(k))1T(t'a(k+1) ... V0(k+l))1-(Va(k+1+1), - - -Va(k+l+m))-aER

In order to compute the sum over the whole group Sk+l+m, we sum overall the residue classes R and, simplifying the scalar factors, we obtain theequation

(k! l! m!) - (wk A ill) A pm (v1, ... , tk+l+m )

E sgn(a)w(va(1), ... Vo(k))17(Vo(k+1), - - Vo(k+l))1(Va(k+1+1), . Vo(k+l+m))-Sk+I+m

This shows the associativity of the exterior multiplication of forms. The lastformula (5) is proved analogously. 0Definition 2. The exterior algebra A(V') of the vector space V is formedby the sum of all exterior forms

A(V*) = Ek=0

endowed with the exterior multiplication A of forms as multiplication.

Next we will construct an explicit basis of the vector spaces Ak(V*). Todo so, start from any basis el, ... e of V and denote by a, ... , o thedual basis of the dual space V' _ A' (V*). For an ordered k-tuple of indices(k-index for short) I = (i1 < ... < ik) let of denote the k-form defined bythe formula

of := ail A ... A o;k .Obviously, for a fixed k-index J = (j1 < ... < jk),

J 0 if 196 J,ol(ejl, --.,eik) t 1 if I = J.

In particular, the k-forms of are linearly independent in Ak(V'). For di-mensional reasons, this immediately implies

4 1. Elements of Multilinear Algebra

Theorem 3. Let e1, ... , en be a basis of the vector space V, and denote byol.... , o its dual basis in the dual space V. Then the forms o,,1 = (i1 <... < ik), are a basis for the vector space Ak(V*).

Exterior forms can be pulled back under a linear map. In fact, if L : W -+V is a linear map from the vector space W to the vector space V, andWk E A(V') is an exterior k-form in V, then the formula

(L*wk)(wl, ...,Wk) := wk(L(wl),... , L(wk))

defines an exterior k-form (L'wk) E Ak(W`) in the vector space W. Passingfrom the form wk to the induced form L*(wk) is compatible with all algebraicoperations. In particular, the following formula holds:

L' (wk n,/) = (L*wk) A (L5,1)

Furthermore, a vector can be inserted into an exterior form, and the resultis an exterior form of one degree less. Let wk E Ak(V*) be a k-form on Vand vo E V any vector. Define a (k- 1)-form (voJ wk) E Ak-1(V*) by theformula

(voJ wk)(v1, ...,vk-1) := wk(vp,V1, .... Vk-1)

The (k -1)-form vo J wk is called the inner product of the vector vo with thek form wk, and will also be denoted by The antisymmetry of thek-form wk leads to the following relation for (k - 2)-forms:

vl J (vo J wk) = - vo J (vl J wk) .

From now on, let V be a real vector space equipped with a non-degeneratescalar product g. This is a symmetric bilinear form,

g : V x V ---- R,

with the property that the linear map g# : V - V` from V to the dualspace V' defined by

g#(v)(w) := g(v,w)is bijective. For a given basis e1, ... , e,, of V, the matrix

M(g) _ (g(e{, ei))ij-1

is symmetric and invertible. For brevity, its entries will be denoted bygig := g(ei, e,), the entries of the inverse matrix (M(g))-1 by g'j. Recall thefollowing result, which goes back to Lagrange and Sylvester:

Theorem 4. Let g be a non-degenerate scalar product on the real vectorspace V. Then there exists a basis e1, ...,en in V such that the matrix

1. Elements of Multilinear Algebra 5

M(g) is diagonal, i. e.

M(g) =

1

1

0

0

L -1 JThe number p of (+1)-entries as well as the number q of (-1)-entries inthis diagonal matrix are independent of the particular basis. The pair (p, q)is called the signature of the scalar product g, and the number q is called itsindex.

First we extend the scalar product g to the spaces Ak(V*) of k-forms, keep-ing the same symbol for its extension. This continuation is done, relative toan orthonormal basis, by means of the formula

k k = ilil ikik k kg(wl,w2) - g ...g wi(ei,, ...,eikw2(ei ...,eik)it <...<ik

It is not difficult to see that this defines a non-degenerate scalar product in allthe spaces Ak(V') which does not depend on the choice of the orthonormalbasis. The signature, however, changes. For example, if g is a scalar productwith signature (n - 1, 1) in V, then the induced scalar product in the spaceAk(V*) of exterior k-forms has signature ((°A 1), (k-i)) If the original basis

el, ... , en in V is orthonormal, the basis al = ail A ... A aik (i1 < ... < ik)is an orthonormal basis in Ak(V'). Here a1, ...,an denotes the dual basisin V*. Moreover, the length of the k-form al is computed by the formula

g(a/, al) = gu 11 .... g`k`k

In particular, the scalar product in the one-dimensional space An(V') ispositive or negative definite, depending on the parity of the index q of thescalar product.

Apart from a scalar product g, we fix an orientation in the real vectorspace V. Let us recall what this means. Consider the set B(V) of allordered bases B = (vl,...,vn) in the vector space V. For two orderedbases, B = (v1, .. . , vn) and B` = (vi.....v), there exists an (n x n)matrix A(93, 'Z*) = (aij) j=1 such that

n

vi = aijvjj=1


We define an equivalence relation N on the set B(V) by requiring

B N'B' if and only if det(A(B, B')) > 0.Obviously, there are precisely two equivalence classes.

Definition 3. An orientation of the real vector space V is the choice of oneof the two equivalence classes in the set B(V) of all bases in V.

In the plane R2, an orientation can be understood as a sense of rotation:

Figure 1

el elFigure 2

Figure 1 illustrates R2 with the orientation determined by the basis (el, e2),whereas Figure 2 depicts the plane R2 equipped with the orientation (e2, el).The transition matrix between both bases is the matrix

0 1A __ [00

with negative determinant, det(A) = -1. Hence the bases (er, e2) and(e2, el) represent the two possible orientations of R2.

Example 1. Let (Vg) be a real vector space with fixed orientation. Wechoose a basis el, ..., e,, in V in such a way that the matrix M(g) has thediagonal format of Theorem 4, and (el, ... , e,) is positively oriented. Thenthe formula

g(vl,el), ...,g(vn,el)dV(vl, ...,vn) := det

g(vi, en), ... , g(vn, en)

defines an n-form dV E An(V') independent of the choice of the particularbasis e1, ... , e with the stated properties. This form dV is called the volumeform of the oriented vector space with the non-degenerate scalar product g.By means of the dual basis al, ..., an, the volume form is represented as

dV = (-1)Q al A...Aa,,.Here q denotes the index of the scalar product g. The length of the volumeform is, by definition,

g(dV, dV) = (-1)9

1. Elements of Multilinear Algebra

The volume form of the coordinate space R" with the euclidean scalar pro-duct coincides with the determinant:

dR"(vl, ...,vn) = det(vi, ...,vn), v, E R".

In particular, the determinant turns out to be an n-form on the space R".

Now we introduce the *-operator (Hodge operator) assigning a k-form toevery (n - k)-form. Consider the given real vector space V together with ascalar product g and a fixed orientation. For each k-form Sc E AV'), therule

An-k(V*) E) gn-k _ Wk A r)n-k E An(V')determines alinear mapping from An-k(V*) to the 1-dimensional spaceAn(V*). The volume form dV is a basis vector in An(V*), and the vectorspace An-k(V*) is equipped with a non-degenerate scalar product. Hence,there exists exactly one (n - k)-form-to be denoted by *wk-such that

Wk A,fin-k = g(*Wke71n-k)dV

holds for all (n - k)-forms 17n-k. Summarizing we defined a linear mapping

*: Ak(V*) -A n-k(V*),the so-called Hodge operator.

Example 2. As above, let e1, ... , en be an orthonormal basis (g;i = fb;i)of the vector space V representing the fixed orientation, and let al, ... , anbe the dual basis. For an ordered k-index I = (i1 < ... < ik), denote byJ = (j1 < ... < in-k) the "complementary" index, i. e. the ordered (n - k)-index containing the numbers 11, ... , n}\{il, ... , ik}. The equations....g(*aj,oj)dV = alAa = sgn

lI

.

Jn

o'1A...Aan = (-1)°sgn1

I.

Jn

dV

and immediately imply the formula

l...*oj = (-1)Isgn

n(I J gijiI ...gin-yin-4a

The following theorem assembles the main properties of the Hodge *-operatorfor a scalar product of arbitrary signature.

Theorem 5.(1) For each k -form Sc, the twofold application of the Hodge operator

is given by * * wk = (-1)k(n-k)+gwk;

(2) for any k -forms wk, r7k, the following relations hold for the exteriorand the scalar product:

g(*wk, *rlk) = (-1)gg(Wk, rlk), Wk A *,nk = (-1)gg(Wk, rlk)dV .


Proof. To prove the first formula we first compute

= (-1)Qsgn(i...n)9.

,... gi..-kin-k * oJ

2y l...n l...n\_ (-1) 5 1 J 9,0, ...9il-kie-ksgn (I J I g";, ...y;k,koi

_ (-1)9(-1)kin-kioi. /The other formulas are consequences of this first one and the definition ofthe *-operator. We have

(*wk, *rik)dV = wk A *I?k = (-1)k(n-k) * flk Awk

= (-1)k(n-k)9(* * r)k,wk)dV = (-1)99(r/k,wk)dV ,

and this implies the remaining identities. 0Example 3. Let n = 2k be an even number. Then the Hodge operatormaps the vector space Ak (V *) to itself,

*:Ak(V*)-Ak(V*).Moreover,

**talk = (-1)k+gWk,

and hence, in case k + q = 0 mod 2, the Hodge operator has the eigenvalues± 1.

Exercises

1. Let a,, ..., an be a basis of V*, let wl = E aj o;, ril = F_ bi a; be twoarbitrary elements from V*, and let µ2 = c;j o; A aj be a 2-form.

a) Compute wI A ill and, in the case n = 3, explain in which sense theexterior product generalizes the vector product;

b) compute wl A µ2 and discuss in case n = 3 the relation to the scalar

product.

2. Prove that each 2-form w2 E A2(V*) can be represented as

w2 = a, A a2 + ... + a2r_i A a2r

for a certain basis a,, - .. , on of V* - Prove, moreover, that the number r isindependent of the choice of the basis and is characterized by the condition

(w2)r 76 (w2)r+1 = 0 .

Exercises

3. Prove that k linear forms ol, ... , ok on V' are linearly independent ifand only if

olA...nok # 0-

4 (Cartan's Lemma). Let o1i ... , ok be linearly independent linear formsfrom V' and let µl, ... µk be arbitrary elements of V. If o1 A pi + ... +ok A Ilk = 0 holds, then the forms pi are linear combinations of the off,

k

J=1

Moreover, ail = aji.

5. Let e1, .... e,a be a basis of the vector space V and o1, ... , o the cor-responding dual basis. Then the following formula holds for every k-formWk:

n

o,A(eiJ Wk) =i=1

6. For a given form 0 96 wk E Ak(V*), define the subspace M(Wk) C V' by

1tf(wk) = {171 E A'(V*) : ql AWk = O } .

Prove the following statements.

a) The inequality dim M(wk) < k holds:b) the equality dim M(wk) = k holds if and only if there exist k linear

forms Cl, .... ok such that wk = of A ... A ok.

Forms of the kind wk = of A ... A ok are called decomposable.

7. Prove the following statements.

a) A 2-form w2 = F_i.j wig oiAaj is decomposable (see the previous exercise)if and only if

WijWpg - WipWjg + WigWjp = 0;

b) a 3-form w3 = Eij k Wijk of A of A ok is decomposable if and only if

WickWpgr - WijPWkgr + wi.jq Wkpr - wilt wA-M = 0-

8. The Hodge operator of a 4-dimensional vector space V maps the spaceof 2-forms into itself,

* : A2(V') -A2(V'),


If the index of the scalar product is even, q = 0, 2, then the Hodge operatordecomposes the real vector space A2(V') into the eigenspaces

A2 (V') = {w2 E A2(V') : * W2 = f w2

In the case of an odd index, q = 1, 3, the complexification A2(V') $ Canalogously decomposes into the ±i-eigenspaces. Compute the dimensionof the eigenspaces in both cases, and determine a basis of eigenforms.

Chapter ,2

Differential Formsin '

2.1. Vector Fields and Differential Forms

Vectors in euclidean space R" can be understood as "free vectors" or as"place-bound vectors" at a point in R". In the first case, we simply considerR" as a euclidean vector space. The second point of view is based on theconcept of R" as a set or a metric space whose elements are called the pointsof space. The "place-bound vectors" then form vector spaces of their own,each one consisting of all those attached to a particular point. For example,vectors at different points cannot be added. This second concept of a vectorin R" leads to the notion of the tangent space at a point in R".

Definition 1. Let p E R". The tangent space to R" at the point p is definedto be the set

TpR" :_ {(p, v) : v E R" } .

An addition and a multiplication by scalars \ E R are introduced in thisset by (p, v) + (p, w) := (p, v + w) and A (p, v) :_ (p, \ v), respectively.These operations endow each TpR" with the structure of a real n-dimensionalvector space.

The ordinary differential of a smooth map f from the space R" to the spaceR'" at the point p, D fp, can now be interpreted as a linear map f.,p betweencorresponding tangent spaces.

Definition 2. Let U C R" be open and f : U R' be a differentiablemap. For each point p in the set U, the linear map f.,p : TpR" -. Tf(p) Rm

11

12 2. Differential Forms in 1R"

is defined by the formula

(f (p), Dff(v))

Definition 3. A vector field defined on an open subset U of IR" assigns avector V(p) E TDR" in the corresponding tangent space to each point p E U.

If e1.... , e is the standard basis in euclidean space lR". the vector fielddetermined by p (p,ei) is usually denoted by 8/8x', i.e. (8/8x')(p) :=(p, e,). (Compare Exercise 5 for an explanation of this notation.) Obviously,every other vector field defined on U can be represented in the form

V (P) = V, (P) iix-l0 (P) + ... + V" (p) 81" (P)

with certain functions V1..... V" defined on U. The vector field V is calleddifferentiable of class Ck if all its component functions V1, .... V" have con-tinuous partial derivatives up to order k. The set of all vector fields of classCk is, on the one hand, a real vector space and, at the same time, a moduleover the ring Ck(U) of all real-valued Ck-functions on U. Graphically, avector field can be depicted by drawing at each point p the correspondingsecond component v of the vector from the tangent space. Consider, e.g.,on the plane R2 with coordinates x, y the vector field

V(x, y) =

The following figure depicts this vector field:

yA

ttt

ttTf

Each of the tangent spaces Tp R" is a real vector space. Hence we canconsider the dual space, 71R' := (TpR, "")'. as well as its exterior powers,

/"(R") := "k(T;R")

2.1. Vector Fields and Differential Forms 13

An element wk of the space Ap(R") is thus an antisymmetric multilinearmap with k arguments on the tangent space TplR":

wk : Tp1R" x ... x TpR" -p R.

In analogy with the notion of vector field, we will now introduce the notionof differential form.

Definition 4. A k -form on the open subset U of lR" assigns to each pointp E U an element wk(p) E Ap(1R")

First we want to consider some examples of differential forms.

Example 1. Let f : U R be a smooth real-valued function, let p E U bea fixed point, and let D fp : IR" -, IR be its differential at the point p. Thenthe formula

df (p)(p, v) := Dfp(v)obviously defines a 1-form df on the set U.

Example 2. A fixed basis el, ... , e" in IR" determines n coordinate func-tions x1, ... , x" and hence their differentials dx', ..., dx". Thus for a tan-gent vector (p, v) E TpR" we have the identity

dx`(p)(p,v) = v`,

where v` denotes the i-th component of v with respect to the basis eI, ... , e".In addition, the 1-forms dxl (p), .... dx"(p) form a basis of the vector spaceA' (R") = T R". Arbitrary exterior products dx" A ... A dx'k as well astheir linear combinations with functions as scalar coefficients lead to furtherexamples of k-forms. Conversely, each k-form wk on U can be representedas

k = 11 ikw - wjl....,tkdx A ...ndx;1 <...<ik

with certain functions w;,,

A differential k-form of class C' is defined as a form wk whose coefficientfunctions w;,,,.,,;k all are of class C'. The set of all these forms is denoted bySti (U). Obviously, this is a real vector space as well as a module over thering C'(U). The 0-th exterior power of a vector space is its field of scalars.Therefore, the 0-forms of class C' are taken to be the C'-functions,

1l?(U) := C'(U).

Example 3. Let f be a real-valued function of class C' on the open setU C IR". Then df is a 1-form of class C". and

df = 209X LI dx'+...+'f dx".

14 2. Differential Forms in R"

In fact, at the point p E U the following equality holds for the vector (p. v):n of

df(p)(p,v) = Dff(v) _ (P)v'axi

Replacing the vector components v' by dx'(p)(p, v) and omitting the argu-ment (p, v), we arrive at the stated formula

n 9fdf (p) _ (P) . dx' (p)

The exterior product of multilinear forms can easily be extended to differ-ential forms, defining for two forms wk, 7771 on U a (k + l)-form by

(wk An`)(P) := wk(P) A7l(p)

The rules known from the first chapter remain valid without change:

(1)(wk+µk)A7j' = wk AY/ +µkA7/';

(2) (f wk) A = f wk A i/;(3) wk A ii = (-1)kl7! A wk.

In particular, the lastproperty

implies that the exterior product of a formof odd degree with itself always vanishes, e.g., dx` A dx' = 0. Forms of evendegree do not in general have this property: A straightforward calculationshows that for the forms on R4 below the following relations hold:

(dx' A dx2 + dx3 A dx4) A (dx' A dx2 + dx3 A dx4) = 2 dx' A dx2 A dx3 A dx4 .

For conceptual as well as computational reasons, it is important that differ-ential forms can be "pulled back" by maps.

Definition 5. Let f : Ul --y U2 be a differentiable map between two opensubsets Ul C R" and U2 C R, and let, moreover, wk be a k-form on U2.Then a k-form f' (wk) (the pullback or induced form) on Ul is defined by

f,(wk)((P,Vi),...,(p,vk)) := wk(fa,P(P,vl),...,f..P(P,vk))The form f * (wk) obeys the following rules.

Theorem 1. Let f : Ul -, U2 be a differentiable map between the open setsUi C R" and U2 C R' with component functions f'. Then

f`(dx`) _ d,,," 8xi

Let, moreover, differential forms on U2 and a function g : U2 -p R be given.Then

(1) f*(wi +w2) = f*(wk)+f*(w2);


(2) f' (g . wk) = (g ° f) f *(wk);

(3) f*(wk Arl1) = f'(wk) A f*(r/).Furthermore, in the case n = m, the following additional equality holds:

Proof. The definition of f' implies

f*(dxt)(P)(P,v) = dx'(f(P))(f(P), Dfp(v))

But this is precisely the i-th component of Dfp(v). Hence

4(p)dx'(p)(p,v).f(dx')(P)(P, v) = at (P)zai=1 i=1

The three rules immediately follow from the definitions and will thereforenot be proved here. To derive the last equation, we use the second rule andthe identity just proved:

A...ndxn) = (guf)f*(dxl)n...A fs(dx")

= (g f)afdxJI] fn n

°A ... A

8xiît=1 in=1

nn afl (9f_ (g ° f) E aTi1 " ' 5; dxit n ... A dxinit, ....i.,=1

Since the exterior square of a 1-form always vanishes, in this sum onlythose terms remain whose n-index (j1.... ,j) is a permutation of (1, ..., n).Using the antisymmetry of the multiplication in the exterior algebra, weobtain the formula

(g°f)Esg11(1 J n)BxJi'...'8xj. dx1 n...ndx"J

and thus the determinant of the differential D f . 0

Recall that a Ci-function f on U can be considered as a 0-form, so thedifferential d is also a map turning a 0-form f into a 1-form df. Applyingthis map d in a suitable way to the coefficient functions of a differentialform, the exterior derivative can be extended to act on k-forms in general.

Definition 6. Let wk be a k-form on the open set U,k E it

Aik

wit, .ik dx ... A dxit <...<ik

16 2. Differential Forms in IR'

We define its exterior derivative dwk by the formula

CO = E d(wi1....,ik) dxi' A ... A dxiki 1 <...<ik

nwi1.... A. a n dx'i n . . . A dxk:.axa dx

it <...<ik a=1

Hence dwk is a (k + 1)-form of class Cl-1, and d becomes a linear operatorbetween corresponding spaces,

d: nk(U) Qi+1(U).

Example 4. Consider on R2 the 1-form

w1 = sin x dy + sin y dx .

Since d(sin x) = cos x dx + 0 dy and d(sin y) = 0 dx + cosy dy, we have

dw' = cos x dx n dy + cos y dy n dx = (cos x - cos y) dx n dy.

Theorem 2. The exterior derivative obeys the following rules:

(1) d(wk +qk) = dwk+dqk forwk, 17k E Qk(U);

(2) d(wk n 171) _ (dwk) Aril + (-1)kwk n (drll);

(3) d((Iwk) = 0 for wk E s22(U);

(4) f*(dwk) = d(f*wk) forwk E f21(U2) and f : U, - U2 c R"'.

Proof. The first identity is trivial. For the second, we use the multi-indexnotation ! = (i1 < ... < ik) and set dx1 = dxi' n ... A dxik as well as

wA = > wl dxl , rlt = > qj dxJ .

ThusI J

wk A rli = wlr,J dx' n dxJ ,I,J

and, applying the exterior derivative, we obtain the formula

d(wk n q1) E (8x° qJ + wIa7oI dx° n dx1 n dxJ .

a=

The first summand equals dwk A7/, and in the case of the second preciselyk transpositions lead to

d(w'nr71) = dWknr7'+(-1)kEEwI" dx'ndx"ndxJI,J a=1

= dwkAr71+(-1)kwkndr/.


Now we will show that dd = 0. Applying the derivative twice yields theexpression

n dx° n dxiddwk = EE 013x°3

I 3=1 o=1

( a2à,'\ dx3 A dx° A dxl.I a<3 8x3x° ax°x3

By assumption, the second partial derivatives of the functions w/ are con-tinuous. Hence, Schwarz' lemma implies that the expression in bracketsvanishes f o r each multi-index I. Finally, w e show that pulling back differen-tial forms commutes with the exterior derivative. T o this end, let y1, ... , yi1be the coordinates on V, let the form wk = Ei wi dy' be expressed in these.and let f 1, .... f' be the components of the map f : U1 - U2. From

dwk = [ ` wI dy° A dy'

Io=11

we obtain, for the pullback of the form,

f'( k) = 1: ! ww (f(x))df°ndf'1 A...Adf'k./ o=1

y°

On the other hand,

f*wk = w,(f(x))df'1 A...Adf`k,I

and its derivative is computed as follows:

d(f'wk) =n aw,axa(f(x))dx3ndj'1 n... ndf

13=1

(f (x)) af- dx3 A df" A ... A df'k/ Q=1o=1

Making use of df° = E(af°/0x3) dx,3, this can be simplified:

d(f*wk) = .EaW°(f(x))dfoAdf'1 A...Adf'k = f*(dwk).I °=1

Example 5. The following example illustrates that, for computational pur-poses, the rules for f * and d formulated in Theorems 1 and 2 often suffice.Consider, e.g., the map

f : R2 - R3, f (U, v) = (u2, v3, uv)

18 2. Differential Forms in Rn

and the 1-form on R3 defined in the coordinates x, y, z by

w' = ydx+xdy+xyzdz.Compute f'(w') as follows. First,

f*(wl) = (yof)f'(dx)+(xof)f'(dy)+(xyzof)f*(dz)= v3 f' (dx) + u2 f' (dy) + u3v4 f' (dz) .

Then we use the fact that the exterior derivative commutes with the inducedmap:

f'(dx) = d(f'x) = d(xof) = d(u2) = 2u du.Similarly,

f*(dy) = 3v2dt,, f'(dz) = vdu+udv,from which the result follows:

f* W) = (2uv3+u3v5)du+(3u2v2+u4v4)dv.

2.2. Closed and Exact Differential Forms

From the theory of the Riemann integral, it is well-known that every con-tinuous function on R has a primitive function. In the language of forms,this can be expressed by saying that for every 1-form µ' = g(x) dx with acontinuous coefficient function g : R - R, there exists a function f such thatdf = p'. If, in addition, g is differentiable, then certainly dµ' = 0, sinceeach 2-form on R vanishes. Now we want to pose the analogous question ofwhether any differential form has a "primitive form":

Let pk E fli (U) be a k-form. What are the conditionsguaranteeing the existence of a (k-1)-form gk-' E ft2 ' (U)whose exterior derivative coincides with µk, dgk-1 = pk

The equality dd = 0 immediately implies that dpk = 0 is a necessary con-dition, but, in general, it is not sufficient for the solvability of the equationdq ' =,[k.

Definition 7.(1) A k-form wk E f1k(U) is called closed if dwk = 0;

(2) a k-form wk E fZi (U) is called exact if there exists a (k - 1)-formnk-' E f4-'(U) such that de-' = wk

The property dd = 0 states that each exact form is closed.

Example 6. Consider on the open set U = JR2-10) the winding form

i -y xW = x2+y2dx +x2+y2dy

2.2. Closed and Exact Differential Forms 19

and calculate its derivative dwi:-dy y(2x dx + 2y dy)1 r dx x(2x dx + 2y dy) l y

Ax2 + y2 + (x2 + y2)2 J dx + [x2 + y2 (x2 + y2)2 A d

- y2+x2-2y2+x2+y2-2x2dx Ad = 0.(x2 + y2)2 y

Hence, wl is closed; later we will see that wl is not exact (Example 10). Theintegral of the winding form along a closed curve surrounding zero measureshow often it "turns around" the origin (Exercises 1 and 2).

In algebraic topology, it is common to describe the difference between closedand exact forms by the so-called de Rham cohomology. The vector spacesof "cycles" Zk(U) and "boundaries" Bk(U) are defined by

Zk(U) {wk E !1k (U) wk is closed},

Bk(U) {WA E S1;(U) WA is exact} .

Thus Bk(U) is a subspace of Zk(U), and the k-th de Rham cohomology ofU is defined as the quotient space

HDR(U) := Zk(U) / Bk(U).

The winding form, e.g., is a non-trivial element in H)DR(R2 - {0}) # 0, andlater (Exercise 14) we will show that

HDR(II 2 - {0}) = R.

The k-th de Rham cohomology only depends on the topological shape of theset U. For example, the de Rham cohomology of a convex set vanishes. Forslightly more general sets this is the contents of Poincare's lemma, whichwe are going to discuss now.

Definition 8. A subset U of IRn is called star-shaped (or a star-region if itis open in addition) if there exists a point po E U with the property thatfor every second point x E U the segment joining po with x is completelycontained in U. Obviously, star-regions are path-connected.


Theorem 3 (Poincare's Lemma). Let U be a star-shaped open set in lid".Then

HDR(U) = 0

for every k = 1, ... , n.In other words: For each closed k form wk E 1l (U) there exists a (k - 1)-form rik-1 E Qk-1 such that drik-' = wk.

Proof. In order to show this, we assign to every k-form wk = wi dx' a(k - 1)-form P(wk) satisfying the identity

wk = P(dwk) + dP(wk) .

For closed forms the first term vanishes, which proves the assertion. Theform P(wk) is defined as follows:

P(wk) _k rj'tk_1wI..ik(tx)dt] _

(-1)°-1I x° dxi' A... Adxio A... A

1j< ... <ik°.

Here the notation dxi° is intended to indicate that the corresponding factoris omitted. Hence, P(wk) is of degree one less than wk, as claimed. Letus compute the exterior derivative of P(wk). To do so, apply the productrule to the coefficient functions: The integral has to be differentiated withrespect to x implicitly, whereas the derivative of xi° each time yields theterm

dxi° Adxi' A...Adxi° A...Adxik.

The sign (-1)°-1 is chosen in order to render this expression equal to dx1' A... A dxik. Altogether, we thus obtain

dP(wk) = k JtlC_twjl..ik(tx)dt dx" A ... A dxikl1<...<ik o

n.k[]tk0';1k(tx)dt]+ L (-1)°-1 xi°dx`jAdxi1A...Adxi°A...Adxik.

;.°=1,11G..<tk

Calculating, on the other hand, dwk as usual leads to

dwk -n

awîi...ik dxj A dx" A ... dxik .it <...<ik 7=1

2.2. Closed and Exact Differential Forms 21

We apply the operator P to this result:

[oma

P(dwk) = E L [ f't tk&i1...ik (tx)dt] xj dxi' A ... A dxikaxi

i1<...<ik.i-1n,k

ll- -1-1 [J'tyk (tx)dt] x°dx Adxi'A...AdxioA...Adx.i1 <.<ik j,a=1

P(k) + dP(wk) = k [1'J tk-lwi1...ik(tx)dt] dxi' A ... A dxik

i1<..<ik 0

n f1 1

+[J

tkaax

ik (tx)dtJ x; dxi' A ... A dxikY- -7i1 <...<ik .7=1 0

fo

1 d /

[dt (tkwl(tx)) dt] dxl

_ > (wj(x) - O . wl(0)) dxl = >2wl(x)dx'.

Poincare's lemma is but an existence result for the desired form 77k-1, anddoes not claim its uniqueness. For example, if 77k-1 is a solution of theequation d77k-1 = wk, the sum 77k-1 +dCk-2 also solves the equation for any(k - 2)-form k-2. Conversely, for any two forms ii-i and ,jr' satisfying

d77i-1 = d772-1 = wk, we have

d(771-1- 772 1) = 0.

Then, by Poincare's lemma, there exists a (k-2)-form do-2 such that ii -1-

772-1 = duo-2. In other words,

77i-1 - 772-1 +

and we conclude that the general solution can always be expressed as thesum of a particular solution and the derivative of an arbitrary (k - 2)-form.

Example 7. Consider on 1R3 the closed 2-form

w2 = xydxAdy+2xdyAdz+2ydxAdz.

We will determine, in two different ways, a 1-form whose derivative coincideswith w2. Let us first explain the Ansatz method to be used. The 1-form 771will be taken as

771 = f (x, y, z) dx + g(x, y, z) dy + h(x, y, z) dz,

22 2. Differential Forms in IR"

where the functions f, g, h : R3 R still have to be determined. Theexterior derivative is easily computed:

d'l' [ax ay] dx A dy + [82 az] dx A dz + I ah - ag] dy A dz.y

Hence the functions have to satisfy the conditions

ag of Oh - Of2 and

ah - OgX Y, 2x

OX ay y' az 49Zy anay

Oz

Integrating, e.g., the first two with respect to x yieldss

dx.g = 2 x2y + ,l dx, h = 2xy + , J49Z

Inserting the result into the last condition, /

2x = 2x + / Oy dx -J

a N dx,

we see that it is satisfied for function f; in particular, we may choosef = 0. Then g = x22 y, h = 2xy, and hence, using

171 = 21 x2ydy+2xydz,

we obtain a solution, as is easily checked. The integration method computesthe "primitive form" by means of the map P(w2) introduced in the proof ofPoincare's lemma. In the example, this turns out to be the sum of 6 terms,namely

1

P(w2) = + Cf

t(tx)(ty)dt J xdy - (J1 t(tx)(ty)dt) ydx

(1o / ro

l+2

1

t(tx)dt J ydz - 2 (J 1 t(tx)dt I zdy

+2 Uo t(ty)dt I x dz - 2(jI

t(ty)dt I z dx.

Each of the summands can easily be computed:

2- 1 2 I 2 2 2 2P(w) = 4x ydy- 4xy2dx+ 2xydz - 2xzdy+ 2xydz - 2yzdx

1 2 2 2 2 4(4xy + 2yz dx + 4x y- 2xz dy + 2xydz.

Which of these two methods leads to a solution more quickly depends onthe particular situation; since an integration has to be carried out in anycase, it might not be possible to find an explicit elementary solution (just.as not every continuous function is elementary integrable).

2.3. Gradient, Divergence and Curl 23

2.3. Gradient, Divergence and Curl

Each tangent space TpRn of the coordinate space is an oriented, euclideanvector space, and hence there is the volume form

dRn(p)E Ap(R")

as well as the Hodge operator

A (Rn) ---, /\p-k(Rn).

This allows us to associate with every differential form wk of degree k onRn a corresponding (n - k)-form *wk defined by applying the *-operatorpointwise, i.e., at each point p E Rn to wk(p). Consider an orthonormalbasis e1, . . . , en of the space Rn and the corresponding coordinate functionsx 1, ... , x". For a k-form wk = F w1dxr expressed in these coordinates theassociated form is thus determined by the following formula:

*Wk sgn l...

Jn wt dxj.

Here, J = (jl < ... < jn_k) is the complementary multi-index to I = (i1 << ik). The volume form dRn itself is simply

dRn = dx1 A ... A dxn .

In addition, we have the possibility to pass from a vector field V to a 1-formwv, and vice versa. This is accomplished by

Definition 9. For a vector field V the dual 1 -form wv is defined by theequation

*wv := VJ dRn .

If the vector field V = E V'8/8x' is expressed in cartesian coordinates, thenthe corresponding representation for w11 is obtained as follows:

wV = (-l)n-I *(V1dx2A...Adxn-V2dx1Adx3A...Adxnf...)= V1dx1+...+Vndxn.

This transition from vector fields to 1-forms will be used now to introducethe gradient of a function and the divergence and curl of a vector field in aninvariant way.

Definition 10. Let f : U --, R be a C'-function defined on an open subsetU C Rn. The vector field grad(f) associated with the 1-form df is calledthe gradient of the function f. The defining equation for the vector fieldgrad(f) thus reads as follows:

' Wj1r.d(!) = grad(f) J dRn = *df .


In the chosen coordinates we have

Of 0-+ -Of agrad(f) = aT1 art + ... + cle axn

Definition 11. The divergence of a C'-vector field V is the function deter-mined by the equation

d(*;.,) = d(V J dRn) := div(V) - dRn .

The formula

div(V) =n 10

ax=

expresses the divergence of V through its components.

The next theorem contains a few simple properties of this operation.

Theorem 4.(1) Let f and g be C' -functions. Then

grad(f - 9) = f - grad(9) + 9 - grad(f) -

(2) For a function f and a vector field V of class C' the followingidentity holds:

div(f - V) = f div(V) + df (V)

Proof. We only prove the second formula. By definition

div(f V) dRn = d(f (V J dRn)) = df A (V J dRn) + f d(V J dRn)= df A *Wv + f div(V) dRn = [41(V) + f div(V)J dRn.

The last step makes use of the following equation, valid for every vector fieldV and any 1-form n1:

171 A *4 = n1(V) (W.

Definition 12. Let f be a C2-function defined on an open subset U C Rn.The divergence of the gradient of f is called the Laplacian o(f) of thefunction f :

0(f) = div(grad(f)) -

In the chosen coordinates the Laplacian is computed by the formula

o(f) _02f

An immediate consequence of the identities in the preceding theorem is theformula

A(f -9) = 0(f)-9+f -A(9)+2(grad(f),grad(9))

2.3. Gradient, Divergence and Curl 25

In particular, there exists an additional operation acting on vector fields indimension n = 3, the so-called curl. For a vector field V and the correspond-ing 1-form wv in R3, the form *d(wy) again is a 1-form and hence in turndefines a vector field, the curl curl(V). In short, we have the following

Definition 13. The curl of the vector field V is the unique vector fielddetermined by the following condition:

dwy =: Curl(V) J dR3 = *'''curl(V)

The definition of the exterior derivative &-'y

dwy =

av2 _ avl dx1 ndx2+ Iav3 _ aV 1 dx1 ^d3+ [aV3 _ av21 dX AdXa21 Ox a21 a23 a22 ax3

leads immediately to the formula

curl(V) = av3 _ av21 a [aV1 _ aV31 a [aV2 _ aV1 ] aa22 a23 J a21 + a23 all 5x2 + 5X1 aX2

]

The properties of the curl of a vector field are summarized in the followingtheorem.

Theorem 5. If the function f and the vector field V are of class C2 anddefined on an open subset U C R3, then

(1) div(curl(V)) = 0 and curl(grad(f)) = 0;

(2) curl(f V) = f curl(V) + grad(f) x V, where x denotes the vectorproduct in R3;

(3) if the curl of V vanishes, curl(V) = 0, and U is star-shaped, thenthere exists a function f such that V = grad(f);

(4) if the divergence of V vanishes, div(V) = 0, and U is star-shaped,then there exists a vector field W such that V = curl(W).

Proof. The first equations immediately follow from the fact that the squareof the exterior derivative vanishes, dd = 0. In fact, we obtain

div(curl(V)) dIR3 = ddwv = 0and

wcurl(grad(f)) _ *dwgrad(f) _ *ddf = 0

Claims (3) and (4) are special cases of Poincare's lemma. For curl(V) = 0 theassociated form wv is closed. By Poincare's lemma there exists a function fsuch that wv = df. Hence V is the gradient of the function f. Property (4)is proved similarly.

26 2. Differential Forms in It"

2.4. Singular Cubes and Chains

We intend to develop the integral calculus for differential forms, and to doso, we need suitable sets as integration domains. First, in this chapter, wewill only allow for subsets of W' which are higher-dimensional analogues ofparametrized curves, so-called singular chains. The k-dimensional unit cubewill be denoted by [0, 1]k,

[0, 1]k = {(x'....,xk)ERk: 05 x1,...,xk<1).

Definition 14. A singular k-cube in U C R" is understood to be a C'-map

ck : [0,1]k -+ U.

Once and for all we agree to call a map defined on any (not necessarily open)subset of Rk smooth if it has a smooth extension to an open neighborhoodof this subset.

Picture of the 2-cube [0, 1]2 B (x, y)' -* cos(x + 1)2 + sin(y - 3)2

Note that not only does a singular cube consist of the image set of ck, butthe notion also involves an explicit parametrization. For example, in casek = 0 a singular 0-cube is a point in the set U. For k = 1, in general, it isa parametrized curve, but it may also degenerate to a point, which explainswhy these cubes are called "singular". In some situations, it can be helpfulto view a singular k-cube as a cube of some dimension larger than k. Noticealso that the boundary of the image set will, in general, have "comers".

2.4. Singular Cubes and Chains 27

Definition 15. A singular k-chain in U is a formal sum of singular k-cubescik in U with integer coefficients Ii E Z.

l1ci +... +lmCkm =: 3k.

k-chains are added in the obvious way, and with addition as compositionthey form the abelian group of singular chains in U, which will be denotedby Ck(U). For example, the inverse of the k-chain sk is its negative -sk.

Definition 16. The standard cube in 1Rk is defined to be the identity mapof the k-dimensional unit cube

Ik : [0, 1]k -+ Rk, Ik(x) = X.

Now we define a boundary for every k-chain. For each index i between1 and k, we parametrize a part of the boundary of [0, 1]k by maps I(i,o)'Ik.l) : [0, 1]k-1 -+ [0, 1]k which insert the value 0 or 1, respectively, atposition is

Ik k-1 k k 1 k-1 - 1 i-1 i k-1(io)[0,1] flR, riio)(x,...,x )-(x,...,x ,o,x,...,xI(i.l) : [0 1]k-1 Rk I(t

1)(x1k-1) = (XI, i-1 i k-1

The boundary of the standard cube Ik is then taken to be the (k - 1)-chaink

alk (I(i 0) - I(i,1) 1 .

i=1

-1

signs

-1

+1

+1

For an arbitrary singular k-cube ck : [0, 1]k - U C 1R we setk

aCk E(-1)i (ck o l(i 0) - Ck o l(i l) Ii=1

Finally, for a singular k-chain sk = l;cf we define the boundary operatorby

19Sk l;aC;


Example 8. We compute the boundary of the 2-dimensional standard cube12. To this end, we label the 4 vertices by pi to p4 starting at the originand moving counterclockwise, and the 4 edges by sl to $4, again movingcounterclockwise and starting at pl. Then

012 = S1 + 82 + sg + s4,

and the boundary of this chain is

8812 = (p2 - pl) + (Ps - p2) + (p4 - Ps) + (pl - p4) = 0.

As in the case of the exterior derivative d, the square of the boundaryoperator vanishes.

Theorem 6.(1) The boundary operator a : Ck(U) -- Ck_1(U) is a group homomor-

phism;

(2) in the sequence

... 8 Ck+1(U) -' Ck(U) a Ck-1(U) - ... 8 . C1(U) a CO(U)

the equality 08 = 0 holds, i. e., for every k-chain sk E Ck(U) wehave

a(a sk) = 0.

Proof. The property 8(si + 82) = 8s1 + 8s2 immediately follows from thedefinition of the boundary operator. Because of linearity, this obviouslysuffices to prove the second statement for an arbitrary k-cube ck : [0, 1]k

U C R". By definition the boundary is determined by

k \ack = (-1)j (ck o IV.o) ck o Ik 1i 1;_1

2.4. Singular Cubes and Chains 29

Applying the boundary operator again, we obtain (omitting the compositionsigns in the second line)

k / la(ack) = >(l) j 1 a(ck o I(j,o)) - a(ck o Ik 1))/j=1

k k-I

= E j:(-1)i+j [ckI(j o)I(+.oj - C'(j,o)I(i 1) - C'(j,1)I(i:)j=1 i=1 L

In order to be able to transform this expression, we introduce two additionalsummation indices a,,3 = 0, 1, and rewrite it as

k k-1a(ack) _ (-1)i+j+a+0 l(ck o I(j.a) o

j=1 i=1 a=o,10=o,1

For j < i and a,,3 = 0,1 we havek k-1 1 k-2 k 1 i-1 i k-21U-) 01 (i.li) (x , ... , x ) = I(j,a) (x , ... , x , fl, x , ... , x )

= (x1, ... , xj-1,

Similarly, one rewrites I(i+1 p) o 1(j,a) as

1(i+1.0) ° IU*(x1, ... , xk-2) = I(i+1,0)(x1, ... , xj-1, a, x1, ... , xk-2)1 1 i-1 i k-2

Together these yield the identity

I(j,a) ° I(i,8) = I( i+1,0) ° I(j,)

In order to apply this, we split the sum appearing in the expression forO(Ock) into those terms for which j < i and the remaining ones:

a(ack) = L.rLam(-1)i+j+a+Qck o Ik o Ik-1

1<j<i<k-1 a.0

+ E E(-1)i+j+a+1Ck 0 I(j,a) 0 I(i,Q)i<j a,0

In the second sum we replace i by j as well as j by i + 1:

a(8Ck) _ (-1)i+j+a+0ck o ik jk-1(,a)

1<j<i<k-1 a,(3

+ E(-1)i+j+l+a+Ock o j(i+1.a)° I(j,

3)1<j<i+1 a,6= 0,

by the identity proved before. 0


In a similar way as the de Rham cohomology, we can define the k-th cubichomology group of a set U C Rn:

Hk°''(U) := ker (8 : Ck(U) --' Ck-1(U)) / im (0: Ck+,(U) ~ Ck(U))

2.5. Integration of Differential Forms and Stokes' Theorem

Consider a singular k-cube ck : [0, 1]k _ U C R" of class C', as well as ak-fonn wk on U. Then the induced differential form (ck)'wk is defined onthe unit cube [0,1]k, and, as such a multiple of dxl A A dx"

(ck)`wk = f (x) dxl A ... A dx'

for some function f : [0, 1]k - R. We can thus define the integral of wk overthe singular cube ck as follows:

Definition 17. Setwk f(x)

ck [01Jk

and extend this definition linearly to any k-chain sk = >, lj in U by

rwk1kWk

Example 9. For k = 1 a singular k-cube is simply a parametrized Cl-curve c : (0, 1] -i U C R'. In this case, the integral of the 1-form wl =p1dyl +... + p"dy" is called the line integnzl of wl along c. If cl, ..., c" arethe component functions of c, the pullback of the form is written as

c'w'(t) = pl(c(t)). d dtt) dt + ... + pn(c(t)) - dcndtt) dt,

and hence, in this situation, we obtain the following general formula for theline integral:

Jw' =

J1 pi(c(t)).

dctt)1dt.

If the 1-form wl is the differential of a smooth function f, then by

c*(wl) = c*(df) = d(f o c)

we obtain the following value for the integral over df :

Jdf = jdt)dt =

In particular, the line integral of an exact 1-form depends only on the endpoints of the curve and not on its shape. It vanishes for a closed curve.

2.5. Integration of Differential Forms and Stokes' Theorem 31

Example 10. We will use the last remark to show that the winding form,

cal = x2 +2

dx + x2 x 2 dy,Y2 y2

is not exact on R2 - {0}. To this end, consider the parametrization c(t) _(cos 21rt, sin 21rt) of the circle by the interval [0.1]. Then

c'wl = 27r dt and jw1 = 27r.

The integral of a k-form over a singular k-cube is, up to sign. independentof the parametrization of the cube. This is a consequence of a well-knowntransformation rule for higher-dimensional integrals: Let y, : U -' V be adiffeomorphism, and let f : V -+ R be integrable; /then

f(f o )(y) . I det(D(y))I dy J f (x)dx .V

Two parametrizations and 4 : [0, 1]A U of one and the same pointset in U differ by a diffeomorphisin cp : [0,11k [0, 1]k of the unit cube.4 = ci o gyp. The determinant of the differential Dp(x) has constant sign,which will be denoted by e(,p).

Theorem 7. In the above situation we have

Jk= f wkcz"

Proof. By Theorem 1 the pullback of the form

( i)"wk = F*(ci)"(wA) = cp`(f dxlA...Adx") = (foip)'4p'(dx'A...ndx")coincides with

(c2)*wk = (f o cp) dxl A ... A dx".

Thus the statement follows from the quoted transformation rule for n-dimen-sional integrals. 0

In case k = 1, changing the direction of a curve results in a change of signfor the line integral.

Now we prove the essential result of this chapter, Stokes' theorem. This isa far-reaching generalization of the Fundamental Theorem of Calculus andcomprises the classical integral formulas of the 19-th century as special cases.The proof to follow will, however, reveal to the careful reader that the coreof the matter is the fact that integration and differentiation are mutuallyinverse operations.

32 2. Differential Forms in 1R"

Theorem 8 (Stokes' Theorem). Let wk be a differential form defined on theopen subset U C IR", and let sk+l : [0, 1]k+l - U be a (k + 1)-chain. Then

f.9$k+l

wk = r d kJgk+1

Proof. Since the integral is additive, it suffices to prove the formula forsingular (k+1)-cubes. First we consider the standard cubelk+l : [0, 1]k+l

R k+1 and represent the k-form wk on Rk+l ask+l _

dxk+1wl~ = fi d21 / ... / dxi A ... A

The derivative is theni=1

k+lk = [(_1)1_1L]i dxl A ... A dxk+1

i=18xi

Hence, by the definition of the integral,

f k+1

Jdwk

axi dx

jk+1 i-1 [0,11k

On the other hand, applying the maps IV Ql parametrizing the differentparts of the boundary of the unit cube leads to the formula

Ik+l * k- 1,

X j 1 , a, Xj. . . ., xk) dx1 A . . . A dskf,(x( > w - ...,J.a )

from which we conclude that

J(Ii I ` U ) _ f k

[0.1[k

/10111k

` J

r flJ J afi (xl, ..., J-1, t, xJ, ... , xk)dt dxl ... dxko dxj

[o.1[k!

adx.dxi

[O,1Jk+l

By the definition of the boundary of Ik+1, this implies thatkr+l r kr+1

Jk w= L L. f f dfx)'dx =fk

ajk+l J-1 Q-o'1Ik+1 j= l[o.llk+l lk+1

li.Q)

and hence we have verified Stokes' formula for the standard cube. For anarbitrary singular (k + 1)-cube ck+1 : [0, 1]k+1 U C 1R" we now use the

2.5. Integration of Differential Forms and Stokes' Theorem 33

fact that the exterior derivative commutes with the pullback of forms. Wehave

J dWk = f (ck+ll*( k) d((Ck+l)'Wk)

Ck+1 Jk+1 l J jkJ+1l` J J

j=1 a=0,1Ck+lolk+1U.u)

k+1__ k+1 wk _ f

(ck+1 ` Wk

ajk+1j=1 Q=0.1 +1wk

.)k+1

wk = Jk.

ask+l

We already emphasized that the line integral of an exact 1-form is indepen-dent of the particular shape of the curve. As a first application of Stokes'theorem, we will prove that the line integrals of a closed 1-form along twodifferent curves coincide if there exists a continuously differentiable defor-mation of one curve into the other leaving their initial and end points fixed.This leads to the notion of homotopy, which is fundamental in topology.

Definition 18. Two Cl-curves co, cl : [0,11 U C R" are called homo-topic if there exists a C1-map F : [0,1] x [0,11 - U with the properties

F(t, 0) = co(t), F(0, s) = co(0) = c(0).F(t,l) = cl(t), F(1,s) = co(1) = cl(1).

The map F is called a homotopy between the curves co and cl.

Theorem 9. Let co, cl : [0,1] - U be two homotopic C1-curves, and let w1be a closed 1 -form on U. Then the line integrals along co and cl coincide:

Lw1 = jwll

Proof. Choose any homotopy F : [0,1]2 -> U between co and c1. This mapF is, at the same time, a singular 2-cube, and hence the 2-form dw1 can be

34 2. Differential Forms in IR

integrated over F. By assumption we have dwl = 0, and the corresponding

integral vanishes:

dwl.0=IF

On the other hand, by Stokes' theorem, we have for the right-hand side

r dw1 fJ w1 + J _ J _ IJF F (t,O) (1,s) (t,I) (O,s)

These four integrals are computed using the homotopy property of F. First,F(t, a) = ca(t) immediately implies, for a = 0,1,

FlZt,a)w1 = cowl.

Moreover, F(a, s) is independent of s, and hence

Fl*(a,,)wl = 0.

Lastly, these relations combine to yield the equation1 1

cw0 =J

cowl - J l ,0 0

and by the definition of the integral this concludes the proof. 0

We will now generalize these observations concerning line integrals to thehigher-dimensional case. Let sk lj ch- be a singular k-chain withimage set A C IR", and suppose that Oak = 0. The set A can, e.g., be a k--dimensional sphere Sk in R". Consider a smooth map f : A A defined ona neighborhood of A which is homotopic to the identity IdA. This is againsupposed to mean that there is a smooth map defined on a neighborhood Uof A x (0.11 in R"+1

F : A x (0,11 -+ A such that F(a, 0) = f (a) and F(a,1) = a.

ThenFo(skxIdlo,11) :=E

is a singular (k + 1)-chain in R", and, because Oak = 0, the boundary is

OF o (sk x Id10,11) = sk - f o sk.

For a k-form wk defined on an open neighborhood of the set A C W', thefollowing holds.

Theorem 10. If Osk = 0 anldf is homotopic to the identity, then

I k=J wkJsk jock

2.6. The Classical Formulas of Green and Stokes 35

Proof. We compute the difference using Stokes' theorem:

f Wk -J

k = f k = f k

sk fosk OFo(skxId1o,11) Fo(skxld[O,j])

The (k + 1)-form (sk x Id[o,I))`F*(dwk) vanishes. This follows from theimplicit function theorem together with the assumption that A is the imageof a k-dimensional chain. But this immediately implies the statement. 0

2.6. The Classical Formulas of Green and Stokes

In this section, we will discuss the classical two-dimensional special cases ofStokes' formula. Let D C R2 be a subset of R2 which can be representedas the image of a CI-map f defined on the standard cube [0, 112. By ODwe denote the boundary of this set, considered as a singular 1-chain. Thederivative of the 1-form w1 := (x dy - y dx)/2 is the volume form on R2.Hence

fdwlvol(D)2 aD

The formula transforms the calculation of a two-dimensional volume intothe evaluation of a line integral, and this turns out to be a special case ofGreen's formula to be discussed now. Consider the 1-form

wI

and compute its derivative:

dw I = I - -J

dx A dy.

Thus we arrive at Green's first formula.

Theorem 11. Let P(x, y) and Q(x, y) be functions of class CI. Then

I fD

IOQ-8PJdxAdy.D L Ox ay

Suppose now that P(x, y) and Q(x, y) are of class C2, and consider the1-forms

wl P [-LQ dx+ LQ dy] and qI := Q I-5y . dx + 5P dy]

The derivative of the difference wI - 771 is easily computed to be

d(w'-1 ) =Applying Stokes' formula leads to Green's second formula.

36 2. Differential Forms in W'

Theorem 12. Let P(x, y) and Q(x, y) be functions of class C2. Then

[PO(Q) - QO(P)]dx n dy =f [QLP - P l dx + fPQ - QPl dy.

8DJ L 1

Stokes' formula concerning certain surface integrals is, in a similarly simpleway, a special case of the above general integral formula. In fact, let F C R3be a subset of IIt3, which can be represented as the image of a Cl-map fdefined on the standard cube [0,1]2 (a surface piece). By 8F we denotethe boundary of this set considered as a singular 1-chain in JR3. Consider avector field V defined on an open neighborhood of F. Its curl is defined bythe following condition:

dwy := curl(V) J dllt3 = *wcurl(V)

Integrating the 2-form sweurj(V) over the surface piece F, we obtain Stokes'theorem in its classical form.

Theorem 13. Let V be a smooth vector field on a neighborhood of F. Then

J(V1dx1+V2dx2+V3dx3) = / &,;y =

OF F8V2 8V11

1 2 r8V3 8V11 IOV3 8V21 2 3

axl - axe I dx Adx +II axl - 5x3 I dx ndx3+ axe - 'ax-3 dx nda .

F

Remark. Using the volume form dF of a regular surface piece yet to bediscussed, the classical Stokes formula can be written more concisely as

Lcun1'1) dF.Here N denotes the normal vector to the surface in R3. We will return tothis in Chapter 3.

2.7. Complex Differential Forms and Holomorphic Functions

The complexification of the vector space of all real-valued forms is called thespace of complex-valued forms on an open subset of R". Such a form can besplit into its real and imaginary parts;

wk = wo + i Wk

and differentiation as well as integration are defined with respect to thisdecomposition:

d,jk dwo + i dwi, LkjkOLkI .

2.7. Complex Differential Forms and Holomorphic Functions 37

In a similar way we extend the exterior product to complex-valued forms:

W k A 771 := (WO A 170 - W I A 911) + i (,oo A 7l1 + W I A 770) .

Then the previous computational rules and Stokes' theorem still hold. Nowwe want to apply complex-valued forms to study holomorphic functions and,to do so, first identify the real vector space R2 with the complex numbersC. From z := x + i y and z := x - i y we obtain the differential forms

dz := dz := and dzndz =Let f be a complex-valued function with real and imaginary parts u and v ofclass C1, f (z) = u + i v. Denote by u=, uy, vv, vy the partial derivatives withrespect to the corresponding variables. Then f (z) dz is a complex-valueddifferential form. We compute its differential:

Now let f : U --+ C be a complex-differentiable function defined on an opensubset of C. Elementary complex analysis starts by proving that its real andimaginary, parts are smooth functions in the sense of real analysis (Goursat'stheorem). Furthermore, the Cauchy-Riemann equations hold:

ux=vy and uy= -vx.Theorem 14. If f (z) is a complex-differentiable function, then the 1-formf (z) dz is closed,

d(f (z) dz) = 0.

An immediate consequence is Cauchy's theorem.

Theorem 15. Let U be an open subset of C, and let -y be a closed curve inU, which is the boundary of a singular 2-cube. Then the integral

0

vanishes for each complex-differentiable function f.

In a similar way we derive Cauchy's integral formula. To do so, we assumethat f (z) is a complex-differentiable function on a neighborhood of the disc

K(zo,M) = {zEC:IIz-zoII<-M}.Fix 0 < e < M. The function f (z)/(z - zo) is complex-differentiable inK(zo, M)\K(zo, e), and, since d((f (z) dz)/(z - zo)) = 0, we have

f.9K(zo,Al)

f (z) dz = / f (z) dz - f (z - zo) f (z) dzJ JZ - ZO aKi:o,E> Z - zo ax(zo,e) Ilz - zol12

=e2

fK(zo0

(z - zo)f(z) dz.


We compute the derivative of the 1-form (z - zo) f (z) dz:

2i -

Thus

J2i. f(x) dandy.

K(zo,M) Z - ZO

=E2 K(zo.e)

The mean-value theorem of integral calculus states that there is a numberze in K(zo, e) for which

f (z) dx n dy = f (ze)vol(K(zo, e)) = 7r `2f (ze)

K(zo.e)

If e tends to zero, then f (ze) converges to f (zo), and we arrive at Cauchy'sintegral formula:

Theorem 16.f(zo) =

1

27ri JaK(zO,jvf) z - zo

This formula is fundamental for the theory of functions. It implies, e.g.,that every function which is complex-differentiable in the neighborhood ofa point can be expanded into a power series (i.e. is an analytic function).

2.8. Brouwer's Fixed Point Theorem

A fixed point of a map f : X --+ X from a set to itself is defined to be apoint xo which is not moved by f, f (xo) = xo. In topology, several fixedpoint theorems are known. They state that certain continuous maps from ametric space to itself necessarily have at least one fixed point. If, e.g., X isa complete metric space, and f : X -p X is a contracting map, then by theBanach fixed point theorem the map f has at least one fixed point. Thereare, of course, (non-contracting) maps from a complete metric space to itselfwithout fixed points; translations in R" are examples for this. A topologicalspace X is said to have the fixed point property if every continuous mapf : X - X from X to itself has a fixed point. This is obviously a topologicalproperty, i. e., homeomorphic spaces either all have the fixed point propertyor none of them has it. The unit circle X = S' is compact; nevertheless itdoes not have the fixed point property; rotations are continuous maps fromthe unit circle to itself without stationary points. Brouwer's fixed pointtheorem states that the closed n-dimensional ball of radius R,

D"(R) = {xER": IIxii <-R},is a metric space having the fixed point property. We will first prove thistheorem in the case of smooth maps from the ball to itself, and afterwardsextend the proof to continuous maps by means of an approximation argu-ment.

2.8. Brouwer's Fixed Point Theorem 39

Theorem 17. Every Cl-map f : D" -+ D" from the n-dimensional ball toitself has at least one fired point.

Proof. Suppose that f : D" - D" does not have any fixed points. Thendefine the map F : D" Sn-' from the ball to its boundary by assigningto every point x E D" the point of intersection of the ray from f (x) throughx with the sphere Sn-'. The formula for F,

F(x) = x

X - f(x)2

x - f(x) x - f(x)_ IIxI12 + (X,IIx - f(x)II 1-

(X'

IIx - f(x)II> IIx - f(x)II

shows that F is smooth. Moreover. F acts on the boundary of the ball asthe identity, F(x) = x for all x E S. Let Fl.... , F" be the componentsof F. Differentiating the following relation, which is valid for all x E Dn.

n

E(F'(x))2 = 1,i=1

yields

2 (Fiaa*5) Idxi = 0,i=1 ij=1

and hence for each index j

j:Fi(x)8F11'(x) = 0.11T1

i=1

Therefore, the system of equations

8P (x) = 0Ox)i=o

has a non-trivial solution (al, ... , an) = (Fl (x), ..., F"(x)) j4 (0.... , 0).Hence the determinant of the matrix

det 0

vanishes. Now we apply this observationto the differential form

Wn-' = F1 A dF2 n ... A dF"

and conclude that its differential vanishes:i \

&,n-1 = dFlAdF2A...AdF" = det(aa2 )dx'A...Adx" = 0.

40 2. Differential Forms in 1t"

By Stokes' theorem the integral of the form w' 1 over the boundary Sn-1of the singular cube D" is equal to zero:

0 = do-1 = n-1- JD- jn-1

On the other hand, F acts on the sphere S"-1 as the identity; hencewn-1ISn-I = x1 dx2 A ... A dxnlSn-i .

This impliesr

0 = J x1 dx2 A ... A din = J dxl A ... A dx" = vol(D"),n -I Dn

and we arrive at a contradiction. 0

Theorem 18 (Brouwer's Theorem). Every continuous map f : D" Dnfrom the n-dimensional ball to itself has at least one fixed point.

Proof. The proof will be reduced to the case of a C1-map by applying theStone-Weierstrass approximation theorem (see, e.g., [Rudin, 19981, Theorem7.32). The ball D" is compact. Consider the ring C°(D") of all real-valuedcontinuous functions on it, as well as the subring R of those functions whichare the restriction to D" of a C1-function with strictly larger, open domainof definition. Obviously, the subring R contains the constant functions andseparates points. By the Stone-Weierstrass theorem, it is dense in C°(U).Applying this to the components f 1, ... , f" of f, we conclude that for eache > 0 there exists a C1-map

p : D' - R" such that IIf (x) - p(x)II < e for all x E D'.

Consider the renormalized map P(x) := p(x)/(1 + s). Since

IIP(x)II - IIf (x)II 5 IIP(x) - f (x)II < e and IIf (x)II < 1,

we have IIP(x)II 5 1 + e; hence IIP(x)II 5 1. Therefore, P is a map from theball to itself. Moreover, P can be estimated against f :

IIf (x) - P(x)II 5 IIf (x) - P(x)II + IIP(x) - P(x)II 5 e + IIP(x)II 11 - 1 +

< e+(1+e)1+e < 2e.

Summarizing, we have proved that for each e > 0 there exists a C1-mapP : D" D" satisfying for every x E D" the estimate

IIf(x)-P(x)II <2e.Now if the continuous map f : D" -+ D" had no fixed point, the number

xinf IIf (x) - xI I

2.8. Brouwer's Fixed Point Theorem 41

would be strictly positive. Choose for F = µ/2 a smooth approximation pwith the properties stated above; by Theorem 17 this map then has a fixedpoint xo E Dn, for which in turn

IIf(xo) - Axo)II = Ilf(xo) - xoII < iwould have to hold. But this contradicts the definition of µ.

Brouwer's fixed point theorem can be viewed as an existence result for real,non-linear systems of equations. We state a possible application, which playsan important role in Galerkin's method.

Theorem 19. Let g1:... , gn : D' (R) -+ R be continuous functions de-fined on the ball Dn(R) of radius R, and suppose that for all points x =(x1, , xn) E S'-'(R) in the sphere Sn-1(R) the following inequality holds:

n9i(x).xi > 0.

i=1

Then the system of equations

91(x) = 92(X) _ ... = 9n (x) = 0

has at least one solution in D'(R).

Proof. We combine the functions to define a map g : Dn(R) --+ Rn, g(x) :=(91(x), .. , 9n(x)). If g(x) 0 0 holds for all points x E Dn(R), we canconsider the map f : Dn(R) -p Sn-1(R),

f (x) := -R g(x)119(x)11

whose image lies in the sphere Sn-1(R). By the fixed point theorem, f hasa fixed point in S"-'(R). Hence there exists a point xo E Sn-1(R) suchthat

xo = -R 9(xo)119(xo)11

This implies R2 II9(xo)II = -R (g(xo), xo), contradicting the assumption ofthe theorem.

In particular, the assumption of the theorem is satisfied for gi(x) = xi +hi(x) if the functions hi : R' -+ R grow more slowly than linear forms,

Ihi(x)I < Ci 11x11"

Under this condition the sumn n

E 9i(x)xi = R2 + c` hi(x)x'i=1 i=1


behaves like R2 on the sphere S"-'(R) and becomes positive for sufficientlylarge radii.

Corollary 1. The system of equations

hl(x) = x', ..., h"(x) = x"

has at least one solution f o r arbitrary continuous functions bounded by II I'

at infinity.

Example 11. The system of equations x = ' 1 + x2 + y'2. y = cos(x + y)has in l 2 at least one solution, e.g.,

x = 1.2758079, y = 0.14722564.

Example 12. Consider in 1R2 the following system of equations:

91(x, y) = x +e-(s+y)2

= 0, 92(x, y) = y + e-(I-U)2 = 0.The picture below diaplays the graph of the function g, (x, y) x + 92(x. y) yover the set [-2.2] x [-2,2]. It shows that this function is positive on thecircle S' (2) of radius 2. Hence the above system of equations has at leastone solution in the disc D2(2). A numerical computation of the solutionleads to the values

x = -0.303122, y = -0.789407.

Exercises 43

Exercises

1. Let f : 1R2 - {0} R2 - {0}, f (r, 0) = (r cos 0, r sin 0), be the polarcoordinate map on the "punctured" plane. Prove:

a) The winding form satisfies f dx) = d9;

b) the radial form x dx + y dy satisfies f ` (x dx + y dy) = r dr.

2. Consider on R2 - {0} the winding form wl = 114.-1., as well as thefollowing family of curves depending on the integer parameter n E Z:

c : [0, 11 - . R2 - {0}, (cos2ant,sin2nrnt).

Compute the line integral of the winding form along the curve c,,, and con-clude that the curves c,l are not homotopic in R2 - {O} for different valuesof the parameter n.

3. Compute the exterior derivative of the following differential forms:

a) xydxAdy+2xdyAdz+2ydxAdz;b) z2dxAdy+(z2+2y)dxAdz;c) 13 x dx + y2dy + xyz dz;

d) e' cos(y) dx - e' sin(y) dy;e) xdyAdz+ydxAdz+zdxAdy.

4. Consider 1R2n with coordinates x1, ... , x2n and the following differentialform of degree 2:

w2 = dx' n dxn+l + dx2 A dxn+2 + ... + dxn n dx2n .

Prove:

a) The form w2 is closed;

b) the n-th exterior power of w2 is related to the volume form via theformula

w2 n ... n w2 = (-1)n(n-1)/2n! dxl A ... A dx2n .

5. The subject of this exercise is to explain why it makes sense to denotethe vector fields x -+ (x, e;), defined using the standard basis e1, . . . , e,, of


R", by a/8x'. Writing these vector fields for the moment as E;, every vectorfield can be written in the form

n

V =

For each function f : U R on an open set U of R" we define a newfunction. the derivative of f in the direction of the vector field V, by

(V(f))(x) := (Df=)(V(x))Prove the formula

i=1which by omitting the argument f provides the explanation asked for.

6. Prove the following rules for vector fields on )1t3:

a) div(V1 x V2) _ (curl(Vl),V2) - (VI,curl(V2));b) curl(curl(V)) = grad(div(V)) - 0(V), where the Laplacian is to be ap-

plied componentwise to V.

7. Compute the line integral

(x- 2xy)dx + (y2 - 2xy)dyJC2along the curve C = {(x, y) E R2 : x E [-1.1), y = x2}.

8. Compute the line-integral

ICsin(y)dx + sin(x)dy,

where C is the segment joining the points (0, x) and (n. 0).

9. Consider on R3 the differential form w2 = y dxAdy. Determine all 1-forms17 1 = p dx + q dy satisfying dal = w2.

10. Prove the following converse to one of the statements of Example 9: Ifwl is a 1-form defined on the open set U, and if the line integral of wl isindependent of the curve, then wl is exact.Hint. Prove this by explicitly constructing a "primitive function". Inphysics, the vector field V corresponding to the 1-form wy is called con-servative, if it does no work along any closed curve -y. 11 wy = 0. Thus, V isconservative if and only if w1, is exact.

n

V(f) _ 08xi (fl

Exercises 45

11. Consider the singular 2-cube, f : [0, 2ir] x [0, 27r] -r S2 C R3 - {0},

f (u, v) _ (cos u sin v, sin u sin v, cos v),

as well as the 2-form w2 = (xdyAdz+ydz Adx+zdxndy)/r3 on 1R3 - {0},where r = (x2 + y2 + z2)1/2 denotes the distance from the origin.

a) Prove that w2 is closed;b) compute the integral of w2 over f ;

c) conclude from the properties just proved that w2 is not exact on 1R3-{0},and that there is no singular 3-chain c3 in 1R3 - {0} whose boundaryequals f, 0c3 = f.

12. Prove that the integral defines a unique bilinear map

HDR(U) X Hkub(U) R, ([WI], [s']) --J

Wk.gk

13. Let w1 = f (x) dx be a 1-form on the interval [0, 1] with f (0) = f (1).Prove that there exist a real number p and a function g with g(0) = g(1) bymeans of which wl can be written as

1W = pdx+dg.

14 (Continuation of 13). Let 771 be a closed 1-form on R2 - {0} and wlthe winding form. Prove that there exist a number p as well as a functiong : 1R2 - {0} -+ R for which

77 1 = pwl +dg.

Consequently, the winding form is the generating element of the first deRham cohomology of R2 - {0}.Hint. Consider the polar coordinate map f from Exercise 1 and its pullbackf `wl. This can be written as f `wl = A(r, O)dr + B(r, B)dO; here B(r, O)dOis a 1-form on [0, 27r] (depending on the parameter r) to which the previousexercise applies.

15. Consider on 1R3 the following exact differential form known from 7:

W2 = xydxAdy+2xdyAdz+2ydxAdz,and the upper half-ball A C S2:

A = {(x,y,z)EIR3: x2+y2+z2=1, z>0}.Prove that the integral of w2 over A vanishes.

46 2. Differential Forms in R°

16. Let C be the circle in R2 with the equation x2 + (y - 1)2 = I in itsstandard parametrization. Compute the line integral

f xy2 dy - yx2 dx

a) directly;

b) using Green's formula.

17. Let E be the ellipse with the equation x2/a2 + y2/b2 = 1 (a > 0, b > 0)in its standard parametrization. Compute by means of Cauchy's integralformula the integral

drJE zand obtain from this the value of the integral

12 dtJp a2 cos2(t) + b2 sin2(t)

18. Let 7-1 be an infinite-dimensional Hilbert space, and D = {x E lI IxI I < 11 its unit ball. Does D have the fixed point property?

19. A subset A C X of a metric space X is called a retract of X, if there isa continuous map r : X --. A such that r(a) = a for all points a E A. Provethat if X has the fixed point property, then so does every retract A of X.

zz

20. The set A = { (x, y) E [-1,1]2: xy = 0 } has the fixed point property.

Chapter 3

Vector Analysis onManifolds

3.1. Submanifolds of Rn

In Chapter 2 we introduced an integration method for differential forms oversets which can be represented as images of singular chains. These sets, how-ever, may be quite irregular, and it is rather difficult to develop a differentialcalculus for functions defined on them. Further notions like tangent space,vector field, etc., are not available either in their context. Hence we will nowrestrict the possible subsets of Rn to a class for which a differential as wellas an integral calculus can be established in a satisfactory manner. Thesesets are called manifolds, and they are-intuitively speaking-characterizedby the fact that their points can be defined locally in a continuous (differen-tiable) way by finitely many real parameters, that is, locally these sets lookjust like euclidean space. It was the fundamental idea of B. Riemann in hisHabilitationsvortrag (1854) to introduce the notion of a manifold as the newbasic concept of space into geometry. In physics, manifolds occur as config-uration and phase spaces of particle systems as well as in field theory. Theprecise description of what a submanifold of euclidean space is supposed tobe is the content of the following definition.

Definition 1. A subset M of Rn is called a k-dimensional submanifoldwithout boundary if, for each point x E M, there exist open sets x E U C R"and V C R' as well as a diffeomorphism h : U V such that the imageh(U n M) is contained in the subspace IRk C R :

h(UnM) = Vn(IRkx{O}) = {yEV: yk+1- =yn=0}.

47

48 3. Vector Analysis on Manifolds

The set U* := UnM together with the map h' := hlu.: U* -- V' := VnRkis called a chart around the point x of the manifold. The sets U* and V'are open subsets of M and Rk, respectively (see next page).

V

Rn

v n (Rk x {o})

h-1

A family of charts covering the manifold M is called an atlas. The no-tion "diffeomorphism" can be understood in the sense of an arbitrary C'-regularity (1 > 1). Correspondingly, we have manifolds of regularity classCL. For simplicity, we suppose in this chapter that all maps, manifolds, etc.,are of class CO°, and for this reason we will simply talk about smooth maps,manifolds, etc. Note that, without any change, all the statements also holdassuming only C2-regularity.

For any two charts (h', U') and (hi, Ur) of the manifold for which the in-tersection u* n u; is not empty, one can ask how they are related. Themap

h'o(hi)-1:

is called the chart transition from one chart to the other. The sets h1(U` nUl) and h' (U' n Ul) are open subsets of the coordinate space Rk, and thetransition function h' o (hi)-1 is obviously a diffeomorphism.

The notion of dimension also needs to be explained for manifolds. If a sub-set of Rk is mapped homeomorphically onto an open subset of the space R,the dimensions of both coincide, k = 1. Under the additional assumption of

3.1. Submanifolds of 1R 49

differentiability, which will always be made here, this is easy to prove: thedifferential of a diffeomorphism at an arbitrary point is a linear isomorphismbetween the tangent spaces T1IRk and TyR', and this immediately impliesk = 1. The corresponding fact for homeomorphisms is a deep topologicalresult going back to Brouwer (1910). In any case, the number k occurringin the definition of a manifold is uniquely determined and will be calledthe dimension of the manifold. Sometimes we will write the dimension of amanifold as an upper index, i.e., denote the manifold also by Mk.

In the first theorem we will prove that, under certain conditions, subsets of1R' defined by equations are submanifolds. This will give rise to plenty ofexamples.

Theorem 1. Let U C lRn be an open subset, and let f : U - Rn-k be asmooth map. Consider the set

M = {x E U : f (x) = 0} .If the differential D f (x) has maximal rank (n - k) at each point x E M ofthe set M, then M is a smooth, k-dimensional submanifold of 1R' withoutboundary.

Proof. The proof is based on a straightforward application of the implicitfunction theorem. If xo E M is a point from M, then there exist an openneighborhood xo E Uxo C U and a diffeomorphism hxo : Uxo -' hxo(Uxo) CRn such that the map f o hzp : hxo (Uxo) - Rn-k is given by the formula

f oh=o (x1, ...,xn) = (xk+1, ...,x').

This implies Uxo n M = h=o (hxo (Uxa) n ]R' ), and hence the chart around thepoint xo E M we asked for is constructed. 0Example 1. Every open subset U C Rn is an n-dimensional manifold with-out boundary.

Example 2. The sphere Sn = {x E Rn+1 : IIxUI = 1} is an n-dimensionalmanifold. To see this, we consider the function f : Rn+1 -, R defined byf (X) = I Ix1I2 - 1. Then we have D f (x) = (2x', ..., 2xn+1), and the rank ofthe (1 x n)-matrix D f (x) on Sn is equal to 1. Theorem 1 implies that Sn isan n-dimensional manifold.

Example 3. Consider a smooth map f : Rn ]Rm and its graph

G(f) = {(x, f (x)) : x E Rn} C 1Rn+m

G(f) is the zero set of the map 1 : IRn+m = lRn x 1R- 1Rm defined byO (x, y) = f (x) - y. The differential D4 has maximal rank equal to m ateach point. Therefore, G(f) C Rn+m is an n-dimensional manifold.


Example 4. The torus of revolution is the surface in R3 described by theequation (0 < r2 < rl )

( x2 + y2 - rl )2 = r2 - z2 .

A parametrization can be obtained by the formulas

x = (r1 + r2 cos cp) cos V,, y = (r1 + r2 cos cp) sin 0, z = r2 sin cp

with parameters 0 < cp, ' < 2ir. The partial derivatives of the functionf(x,y,z) = ( x2+y2-rl)2-r2+z2 are

= 2z,Ox - 2x(1 -21

), of - 2y(121

2), Ozx-+y2 y VI'X +yand it is obvious that the vector D f does not vanish at any point of thetorus of revolution. Hence Theorem 1 applies and shows that the aboveequation defines a manifold.

Example 5. Not every set defined by an equation is a manifold. For ex-ample, consider the set in R2 described by the equation x4 = y2:

Near the point (0, 0) E ]R2 this set is not a manifold. In fact, after havingdeleted this point, any neighborhood of the set splits into four componentsand hence cannot be homeomorphic to an interval. Indeed, the assumptionof Theorem 1 concerning the differential of f (x, y) = x4 - y2 is not satisfied

2at the point (0, 0) E R.

3.1. Submanifolds of R" 51

Example 6. On the other hand, there exist manifolds in R" which cannot bedescribed by systems of equations satisfying the assumptions of Theorem 1.Later we will see that every manifold defined as in Theorem 1 has a particularproperty-it is orientable. An example of a non-orientable manifold is theso-called Mobius strip. One of its parametrizations is

x = cos(u)+vcos(u/2)cos(u), y = sin(u)+vcos(u/2)sin(u), z = vsin(u/2)

with parameters 0 < u < 2ir, -7r < v < 7r.

Apart from equations, manifolds can also be defined by prescribing theirlocal parametrizations (charts). Let us explain this construction principle.

Theorem 2. Let M be a subset of R" and assume that for each point x E Mthere are an open set U, x E U C R", an open set W C Rk, and a smoothmap f : W U such that the following conditions are satisfied:

(1) f(W) = M n U;

(2) f is bijective;

(3) the differential Df(y) has rank k at each point y E W;

(4) f'1 : M n U W is continuous.

Then M is a k-dimensional submanifold without boundary in R.

Proof. For an arbitrary point x E M we choose a map f : W -i U withthe stated properties and denote by y the pre-image of x, f (y) = x. Thedifferential has rank k, and hence we can assume without loss of generalitythat

1rdet 9k 0.

4ak

Consider the map defined by g(a, b) := f (a) + (0, b) with g : W x R"-k -+ R".Then the determinant of the differential of g coincides with the determinantabove, and hence, in particular, it is different from zero. Applying the


inverse function theorem of differential calculus, we obtain two open setsV1 and V2. with (y, 0) E V1 and x E V2 in R", for which g : V1 V2

is a diffeomorphism. We invert this map and denote the resulting inversediffeomorphism by h := g-' : V2 -+ V1. By assumption f is continuous,and hence there exists an open set 0 C R" such that

{f(a):(a,O)EV1} = f(w)no.

Consider now the sets V2 := V2 nO and V1 = g-'(V2). Then we have

V2nM = V2nOnM = {g(a,0):(a,0)EVII,

and thus we obtain

h(V2nAf) = g-1(V2 nM) = V1 n(Rk x 101)

Therefore, the condition to be satisfied for each point of a manifold holdsfor A1.

Now we will extend the notion of manifold, taking into account also bound-ary points. We confine ourselves to the case that the boundary itself is asmooth manifold without boundary (no corners or edges). We define thek-dimensional half space Hlik to be the set

Hk = {xERk:xk>O}.

Definition 2. A subset M of R" is called a k-dimensional submanifold(with boundary) if for each point x E M one of the following conditions issatisfied:

(1) There exist open sets U and V, with x E U C R", V C R", and adiffeomorphism h : U -+ V such that

h(U n M) = V n (Rk x {0}) .

(2) There exist open sets U and V, with x E U C R", V C R", and adiffeomorphism h : U -+ V such that

h(UnM) = Vn(Hk X {O})

and the k-th component hk of h vanishes at the point x, hk(x) = 0.

Conditions (1) and (2) cannot be satisfied at the same time for one andthe same point x E M, for otherwise there would exist diffeomorphismsh1 :U1 -iV1, h2 : U2 --+ V2 such that

h1(UInM) = VlnRk and h2(U2nM) = V2nlEllk, h2(x)=0.

3.1. Subinanifolds of R" 53

The set hl (U1 n U2) then would be an open subset in IRk mapped diffeo-morphically onto h2(Ul n U2) by the chart transition map h2 o h, 1. Sinceh2(x) = 0, the set h2(U, nU2) would thus contain a point from the boundary8Hk = Rk-4 of the half-space. Consequently, it could not be open in Rk.Altogether this contradicts the inverse function theorem. This observationjustifies the following

Definition 3. Let M C 1[t" be a manifold. A boundary point of M is apoint x E M for which condition (2) of Definition 2 is satisfied. The set ofall boundary points is denoted by 8M and called the boundary of M.

Theorem 3. Let AEI be a k-dimensional manifold. Then its boundary OMis either empty or a smooth (k-1)-dimensional manifold without boundary,

88M=0.

Proof. Fix a boundary point x E OM and choose open sets U C R n. X E U,and V C Ht" with

h(U n m) = V n (Hk x (01).

For every other boundary point x' E U n OM the k-th component of h hasto vanish at x` by the preceding observation, hk(x*) = 0. Hence we have

h(U n 8M) = V n (IItk-' x {0}),

and thus (h IunaAf, U n OM) is a chart for the boundary 8M.

Example 7. The boundary of the Mobius strip is a closed space curve.

Example 8. The (n - 1)-dimensional sphere Sn-' is the boundary of then-dimensional ball D".


3.2. Differential Calculus on Manifolds

When a manifold is covered by charts, every chart range is an open subsetof R" and hence a set on which differential calculus is familiar from anal-ysis. In this way it is possible to develop a differential calculus on manifolds.

As in the preceding section, we will now start from a k-dimensional mani-fold Mk and a chart h : U V around a point x and denote by y := h(x)the image of x under this chart map. Then h-1 : V -{ U is smooth and(Dh-')y = (h-1).,y is a linear map between the tangent spaces to R" (com-pare Definition 2, Ch. 2)

(h-1).,y: TyR" - T=R" .

Definition 4. The tangent space of the manifold Mk at the point x isdefined to be the image of TVRk under the map (h-1).,y:

Trllfk (h-1).,y(Ty %) C .,R" .

The tangent space T?Mk is a k-dimensional vector space, since the differ-ential of the diffeomorphism h-1 is injective. Moreover, we have to checkthat the tangent space of the manifold just defined does not depend on thechoice of the chart. But this is an immediate consequence of the equivalentdescription for the tangent space that is to follow next.

Theorem 4. The tangent space TTMk consists of all vectors (x, v) E T=R"for which there exists a smooth curve y : [0, ±E) - Mk C R" such that-y(0) = x and y(0) = v.

Proof. A vector v = (x, v) E TXMk in the tangent space can be representedas v = (Dh-'),(w) for a certain vector w E Rk. The image under h' ofthe straight line in Rk through y in the direction of the vector w is the curve7(t) we were looking for: the equality

-t(t) = h-1(y + tw)

immediately implies -y(O) = h-1(y) = x, and from the chain rule we obtainfor the tangent vector

ddtt) = dt (h-1(y + tw)) l t=o = (Dh-'),(w) = v.

The converse is proved analogously.

If the manifold Mk is defined by (n - k) equations, and if, in addition,the differentials of the defining functions are linearly independent, then thetangent space has a simple description.

3.2. Differential Calculus on ?Manifolds 55

Theorem 5. Let fl, ... , fn_k : 1R" , 1R be smooth functions and supposethat

df1 A ... A 0 0 .Then the tangent space TTMk of the manifold

Mk = {xER": f1(x)=...=f,,-k(:r.)=0}

consists of all vectors v E T11R" satisfying

df1(v) = ... = dfn-k(v) = 0.In particular, the euclidean gradient fields grad(f1), ... , grad(fk) are per-pendicular to the tangent space of the manifold at each point of :11k.

Proof. Taking a curve r) Alk in 11k and differentiating theequation

f1('r(t)) = ... = f"-k(r(t)) = 0with respect to the curve parameter t yields

df1(i(t)) _ ... = df"-k(i(t)) = 0.The tangent space TA1k is thus the subspace of all those vectors v E T IR"on which all the differentials df1, . df"-k vanish. Comparing the dinlen-sions of these two vector spaces shows that they have to coincide.

The set of all tangent spaces to the manifold is called the tangent bundle ofAlk and denoted by TMk. It is a manifold of dimension 2k. In fact, at leastin the case that A1k is determined by equations, f1 = . . . = f"_k = 0, theset TAlk is in turn defined as a subspace of 1R" x 1R" by the equations

fl (:r) = ... _ .fn-k(x) = 0 and df1(x. v) = ... = df"-k(x, v) = 0.

These are 2(n - k) conditions in R2". The corresponding functional determi-nant (Jacobian) does not vanish, since the differentials df1 are linearly inde-pendent. The formula (x, v) := x defines a projection 7r : TMk . Alk onthe tangent bundle of the manifold assigning to every vector its base point.


Example 9. Consider the sphere S' = {x E R"+1 :IIxII = 1}. Thedifferential of the function IIXI12-1 is 2(x1, = xn+1) and hence the tangentspace to the sphere at any point consists of all vectors perpendicular to thispoint:

TS" = {(x, v) E R"+1 x Rn+1: IIxII =1, (x, v) = 01.

Definition 5. Let Mk C R' and N' C R' be two manifolds, and letf : Alk - N' be a continuous map. We call f a differentiable map if foreach chart h-1 : V - Mk of the manifold Mk the resulting map f o h-1V -. N' C R' defined on the open subset V C Rk is differentiable.

As in euclidean space, the differential of a smooth map can be introduced asa linear map between tangent spaces. For a tangent vector (x, v) E TZMk wechoose a curve y : [0, e) - Mk with 7(0) = x and y(O) = v. The compositionf o y(t) is a curve in N1 passing through f (x) E N', and its tangent vectordescribes the result of applying the differential of f to the tangent vector(x, v),

f.,x (X, V) := (f(x), dtf o y(0)I

The differential of a smooth map between two manifolds has all the proper-ties which are familiar from euclidean space.

Theorem 6. The differential f.,2 : TXMC -+ Tj(t)NI of a smooth map isa linear map between the tangent spaces, and the differential of the super-position of two smooth maps f and g is equal to the superposition of theirdifferentials,

(g o f).. = 9.,f1=) ° f*.x

The last formula is the generalized chain rule.

Definition 6. A vector field V on a manifold Mk assigns to every pointx E Alk a vector V(x) E ,,Mk in the corresponding tangent space.

If the map V : Mk - TRn = Rn x R" is smooth, then we will speak of asmooth vector field on the manifold. Vector fields can again be added andmultiplied by functions, so that the vector space of smooth vector fields isa module over the ring Coo(Mk) of all C°°-functions on Mk.

Example 10. The formula V(x) = (x, (x2,-x1,0)) defines a vector fieldon the 2-dimensional sphere (see the figure on the next page).

3.2. Differential Calculus on Manifolds 57

For a chart map h : V Mk C R', which this time and and sometimesalso later will be denoted by h instead of h-1, h.(a/ayi) are vector fieldstangent to Mk defined on the subset h(V) C Mk, and they provide a basisin each tangent space. For simplicity and as long as it is clear to which chartwe refer, these vector fields on the manifold will also be denoted by a/ayi.On the subset h(V) C Mk. every other vector field V can be represented astheir linear combination

k

V(y) _ Vi(y)5ii

i=1y

Here V2(y) are functions defined on the set h(V); using the chart map. nowand then they will also be considered as functions on the parameter set V.These functions are called the components of the vector field V with respectto the fixed chart.

Example 11. In euclidean coordinates on 1R2, consider the vector field

V=x15x2-x25x1

depicted on the next page. Introducing in R2 - (0} polar coordinates by theformula

h(r,p) = (r cos V, r sin so), 0 < r < oo, 0 < <p < 2ir,

we see that V corresponds to the vector field In fact, differentiatingthe map h, we obtain the formulas

ral a a_ 1, 2 5 1(1) ar

h.ar = smV5X2 r 7X1

+ 57X2

49

Cla a _ 2a ,a

(2) h a -rsincp+ rcoscp -xp axl ax2axl +x 57X2


Differentiable functions f : Mk -- R can be differentiated with respect to avector field V. At a fixed point x E Mk we choose a curve ry : [0, EJ Mksatisfying the initial conditions y(0) = x and y(0) = V(x). The derivativeoff at the point x in the direction of V(x) is now defined by the formula

V(f)(x) := tf o y(t)It=o

The result is a C30-function V(f) defined on the manifold Mk. In the nexttheorem we stunmarize the properties of this differentiation:

Theorem 7.(1) (V+W)(f) =V(f)+W(f);

(2) V(f1 + f2) = V(f1) + V(f2);

(3) V(f1 - f2) = V(fl) - f2 + f1 - V(f2);

(4) If the vector field V is represented as V = V'(y)818yi in somelocal chart, then

k

V(f) _ Vi(y)8(foh)i=1

Proof. We will prove (4); all the other claims follow immediately. If thepoint x E Mk corresponds to the point y E V under the chain map h : VMk, then -y(t) = h(y + t(V 1(y), ... , Vk(y))) is a curve in Mk satisfying theinitial conditions -t(O) = x and ry' (0) = V(x). The formula for V(f)(x) to beproved then follows from the chain rule:

k

V(f)(x) = d f o9 (y+t(V1(y),...,Vk(y))) _ V'(y)a(fh). 0

dt i=1'


Next we will discuss the notion of Riemannian metric on a submanifold Mkof euclidean space. The scalar product in R" is denoted by (v, w). Werestrict it to the tangent spaces of the submanifold.

Definition 7. Let M11k C R" be a submanifold. In each tangent space T, Mkthe formula

g= ((x. V), (x. w)) := (v, w)

defines a scalar product. The family {g.,} of all these scalar products iscalled the Riemannian metric of A1k.

In a chart It : V _ Alk, the Riernannian metric is locally described by thefunctions g,j defined on the set V,

a a = ah, ah)gij W = A(y) (ay' ayj

) (ayiami

Thus, the components gij(y) of the Riemannian metric are the scalar prod-ucts of the partial derivatives of the chart map h considered as being vector-valued. When it is clear to which chart we refer, h is often omitted (as inthe examples to follow). The (k x k) matrix (g,3) is symmetric and positivedefinite for each y E V. By g(y) we will denote its determinant, which al-ways is a positive function. Recall that we already agreed in Chapter 1 forbilinear forms to denote by (g`J(y)) the inverse of the matrix (g,j(y)).

Example 12. In euclidean coordinates on R", the entries gij(y) = 6,j areconstant and equal to one for coinciding indices i = j. and vanish in theother cases.

Example 13. In polar coordinates on R2 - {0}, equations (1) and (2)immediately lead to the following coefficients for the usual euclidean metric:

a s _ _ d a_ /a a 29r. _ (ar Or

1,g,-" Or

,8y7

= 0, gw;v a , r .

In particular, in polar coordinates on the plane we have g(r, p) = r2.

Example 14. On the sphere S2 - {N. S} C R3 without north and southpole, we consider spherical coordinates defined by

h(y^, y) = (cos f cos ri', sin, cos t,, sine) , 0 < V < 27r, -ir/2 < z, < r/2.

Computing the tangent vector fields yields

(3)

(4)

a a

a a- Cos sinyaz- axi

+ cos cos y a-2,

a a- sin yo sing

2 + cosy 3 .


For the coefficients of the Riemannian metric on the sphere we thus obtain

g 4_90b-)=cos29 - /(Y0, 9vw= =1.

The determinantt of the metric on the sphere in these coordinates is equalto g(v, cost 0; therefore, the spherical coordinates degenerate at thepoles, which is why we deleted them.

We will use the Riemannian metric g of a manifold in order to associatewith every smooth function a vector field, the so-called gradient. For a fixedpoint x E Mk and a tangent vector v E TIMk we first choose a vector fieldV such that V(x) = v. The assignment v - V(f)(x) determines a linearfunctional TxMk - R on the tangent space, and hence there exists a vectorgrad(f)(x) E TTMk such that the equality

V(f)(x) = gx (grad(f)(x), V(x))holds for all vector fields. The vector field grad(f) is called the gradient ofthe smooth function f : AIk -+ JR.

Theorem 8. Let f : AIk -+ JR be a smooth function, and let h : V _ Mkbe a chart. Denote the coefficients of the Riemannian metric in this chartby gij. Then

k

grad(f) = e(f o h) ij aayi 9 ayi

i j=1

Proof. Inserting the right-hand side into the definition, we computek a(f o h) 9ij a

,a

9 ayi ayi aymij=1

_ E a(f o h) ij k a(f o 11)bi = a(f o h)ayi 9 9jm - L ayi m aym

i,j=1 i=1

According to Theorem 7, the last expression is precisely (8/aym)(f ), whichby the definition of the gradient implies the result.

Example 15. The formula for the gradient of a function defined on an opensubset U c R" of euclidean space expressed in cartesian coordinates is

(5) grad(f) = n Ofaa-i aTi

i=1

Example 16. In polar coordinates on R2 - {0} the coefficients of the metricare

grr = 1, 9rp = 0, gp' = 1

T2


For a function f in the variables r, gyp, the formula of the preceding theoremyields

of rr a of ;, a of a 1 of a(6) (f) = Or g ar + app = ar ar + r2 aap aV

Example 17. In spherical coordinates on S2 - IN, S} we have

g'v;v =

2

95'" = 0, 1,cos y)'

and for the gradient of a function f : S2 - {N, S} -R this leads toOf a

(7) grad(f) = cost _ aV ao a0For example, the height function f (x1, x2, x3) = x3 on the sphere may bewritten in spherical coordinates as f (cp, ') = sin rP, and thus

grad(f) = cos z' .

Hence the vector field grad(f) on the sphere S2 vanishes at precisely twopoints, at the north and the south pole.

Vector fields and the Riemannian metric can be represented in various co-ordinates. Now we derive the transformation formulas for its componentswith respect to different charts. To this end, fix two charts h : V -> Mk andh : V -+ Mk and denote the points from V by y = (y1, ... , yk) and thosefrom V by z = (z1, . . . , zk). The chart transition maps will be denoted by0 = (01, ... , 0k) := h-1 o h and t' = (VG1, ... , aiik) := h-1 o h, respectively.


Theorem 9.(1) Let V be a vector field and denote by V 1, ... Vk its components

in the chart h and by 0, ..., Vk the components in the chart h.Then, for each index I between 1 and k.

k ,V, (Y)

(4(y)) az; (b(y))i=1

(2) Denote by g;j the coefficients of the Riemannian metric with respectto the chart h and by g,; those referring to the chart h. Then theyare related by the formula

k 01 corngi;(y) _ ay;

-,91m (0(y))

l,m= I

In particular, for the determinant of the Riemannian metric thefollowing holds:

2

9(y) = det 1

.

9 (4(y))rNProof. The first formula is a consequence of the chain rule. In fact,

a

h. (a) = ( a)=

k atvr a _ k awl aaz' az' h of az' h` (E ) = E az' a-ax' ayl

This leads to the following formula, describing the vector field in the charth:

k ,a k

\k a

i=1a2

1=1 i-1 ax / aThe second formula follows directly from the first:

a a\ax'' axj

k 811,1 awm a a A atll aij,m2 az' azi C ayl ' aym > = E az' azi1,m=1 1,m=1

if, in addition, the (formally completely equivalent) roles of h and h areexchanged.

We will use these transformation formulas to define the divergence of avector field. This will be taken to be a function on the manifold determinedby a local formula. Therefore, it has to be proved that the correspondingexpression is independent of the choice of the chart.

Definition S. Let V = F; V'(y)a/ay' be a vector field on the manifold Mkexpressed with respect to the coordinates of the chart h : V Alk, and let


g(y) : V R be the determinant of the Riemannian metric in this chart.Define a function div(V) : h(V) R. the divergence of V, by the formula

div(V)(h(y)) 1 ka (v/g V M)

Vg i=1 ayi

Theorem 10. The defining formulas for the divergence of a vector field fortwo arbitrary charts h : V Mk and h : V -- Mk of the manifold Mkcoincide at corresponding points:

1

k

a(,/g-V'(y))= 1

k a(vg= f7 '(z))-azii=1 8yi

The proof uses a formula expressing the derivative of a determinant by itsminors. For the sake of completeness we state it here.

Lemma 1. Let k2 functions hj(x) be defined on an open subset V of lRc.Denote by H(x) := det[hij] the determinant and by A;j(x) the minors of thematrix hij(x). Then

_H(x) _ kOhij Di (x) .

ax'' axr J

Proof. Differentiating the determinant and applying the Leibnitz rule yields

OH(x)ax''

e h12 ... hlk h11 ... hl.k-1 a=te

det +...+det8hk1 hk2 ... hkk hkl ... hk,k-I a8x'

Expanding each of the k resulting determinants with respect to the columncontaining the derivatives leads to the stated formula. 0

Proof of Theorem 10. Applying the preceding formula, we obtain

1 k a( 9(y) V'(y)) = k aV'(y) k ii + V (y) ai (in g(y))

9y) ;=1 ay ;=1 ay i=1 aa/

= L yii

My))i=1 j=1

+ V'(y)ay; In 9(O(y)) +In Idetm

gyri=1


Now we compute the partial derivative of the first sum and simplify theresulting expression, making use of the following formulas:

k aoi at' = al .i azj Jsi=1

This immediately leads to the expression

k- k-1 a_ 9(1/) V`(}/)) = 1 O( 9(z) O(z))

5) ;=1syi

9(z) azi9r-- Vrk 2 i 1 m 1

+ '(4(y)) a )az, i + aziIn Idet ayr I 0

i,j,t=1

Last, we have to show the following equation for each index 1 < j < k:k l

8ai

z1 Oyi + az,j In Idet 00m

TI o

= 0.i,1=1

Using the lemma, the second summand in this equation can be written as

ajln det amm I o V, = (det I o 10

-1 k a(det 00m) OI`

J J-1 k

a2Q° a4,i= (det

Lsyr J o

V,)

syiaya 0° aZ;L J i,a.0=1

The matrix [Ow'/ad] is the inverse of [490m/ay''], and, by Cramer's rule.

This yields

a-j0= D&Q . (det I m 1 0 wl -1

8z°

aIn Idet

aim

I

OP- rk

a20°

az" ay* ayiay,3 azi 19z° .i,°,)9=1

and the equation to be proved becomesk 0n2jpi 00°

kk0-20a atpi 0,03

azjaz° ayi + ayiay'3 azi az° = 0i,° =1

But this identity immediately follows from the fact that p and are mutuallyinverse maps: Differentiate the equation

k n i

ww) a;i=1


with respect to the variables z1, set a = 1, and finally take the sum. 0Example 18. In cartesian coordinates on R", the formula for the divergencebecomes

(8) div(V) _" aVi

axi

Example 19. In polar coordinates on 1R2 - {0}, since g = r, for each vectorfield V = V' 010r + V2a/0' the following formula holds:

1 a(rV') a(rV2) OV1 1 8V2(9) div(V) = r +

a=

Or+ -V1 +

arl V

Example 20. In spherical coordinates on S2, the determinant fo the metricis g = cost ly and thus for every vector field V = V1a/a<p + V2,9/a' thecorresponding formula becomes(10)

1 a(cosip V1) a(cosii - V2) aV1 av2 2div(V) =cos y ( 0p + a ) - acp + a - tan tp V .

Theorem 11. Let V be a vector field and f a smooth function on Mk. Then

div(f V) = f div(V) + V(f).

Proof. It suffices to prove the formula in an arbitrary chart. This is doneby a straightforward computation:

1 8(,g- f Vi) 1 of a(Vfg- y')div(f V) _" i=1

agi"" i=1 '

09 V+ f ay

= f div(V)+EViaf = f div(V)+V(f). Oi=1

The last operator to be introduced in this chapter is the Laplacian 0; itacts on functions and is a second order differential operator.

Definition 9. Let f : Mk --' R be a smooth function. Then the Laplacianof f is defined as a function on Mk by the formula

O(f) := div(grad(f ))

Remark. The operator A : C°O(Mk) C- (Mk) is a linear operator act-ing in the vector space of all C0°-functions on Mk. In mathematics, thereare-concerning the sign of 0-two differing conventions. Classically andmainly in analysis, A is defined as above. On the other hand, in vari-ous branches of geometry and global analysis it is common to introducethe Laplacian via the formula A(f) = -div(grad(f)) and to call this theLaplace-Beltrami operator. There are good reasons to do so, as we will see


in connection with the classical integral formulas. Thus, studying a text,the reader has to be careful to detect which sign convention was chosen bythe author. In this book we decided to use the particular choice of sign forthe Laplacian stated in the definition.

Theorem 12. The Laplacian 0 : C-(A1k) -. C-(Mk) is a linear operator.Moreover, for two functions fl and f2 the following formula holds:

A(fi - f2) = fi ' (f2) + f2 A(fl) + 2 (grad(fl ), grad(f2))

Proof. The linearity of 0 is obvious, and the second formula follows bycombining the corresponding rules for the operators grad and div:

A(fi f2) = div(grad(fi f2))= div(fi grad(f2) + f2 grad(f1))

= fi div(grad(f2)) + f2 div(grad(f1)) + 2 (gr'ad(fi). grad(f2))

0

In local coordinates we immediately obtain from

grad(f)=Ya(foh)i;o

the formula

I k

aa(foh)

AM _ - s i g

Again, we write out the explicit form of the Laplacian in the examples wehave discussed so far.

Example 21. In cartesian coordinates on R",

(11) (OXi)2.=1

Example 22. In polar coordinates on Ilt2,

8 (Of a(rOf

a2f 1 of+

102f(12) o(f) = r ar lrar) + app \ r2 app)) are + r ar r2

Example 23. In spherical coordinates on S2,

(13) fcost 0 OW2 + f,2

-tan fIn these coordinates the height function f (x1, x2, x3) = x3 on S2 can berepresented as f = sin ip. This leads to

A(f) = -2sing(, = -2f,

3.3. Differential Fornu; on Manifolds 67

i. e., f is an eigenfunction of the Laplace-Beltrami operator correspondingto the eigenvalue -2. Other eigenfunctions of A on S2 are constructedstarting from a harmonic and homogeneous polynomial P(xl, x2. x3) in 1R3,0R' P = 0, and restricting this to the sphere S2. The resulting function isan eigenfunction of the Laplacian (see Exercise 25).

3.3. Differential Forms on Manifolds

Up to now, for every manifold Al" we considered its tangent space TAI'and vector fields. Now we go one step further and form the exterior power

AA(T.Mm)

of all k-forms w= : TTM' x ... x T1Mm - IIt at the point x E Alm.

Definition 10. A k-form wk on a manifold /L1m is a family {wi} distin-guishing a k-form wT E Ak(Mt) at each point x E AI'".

The differential of a smooth map f : N" -i M"' between two manifoldsallows to pull back k-forms from Mm to yield k-forms on N". This isaccomplished via the formula

(f'wk)(vi, ... , vk) = wk(f.(vl), ... ,

where vl, .... vk E TyN" are tangent vectors to N' at the point y E N",and f.: TyN" Tf(y)Mm is the differential of f at this point. The k-formf `wk is called the induced form or pullback of wk by the map f on N".This construction can, in particular, be applied to a chart h : V Almof the manifold Alm. Hence, for a fixed chart., to every k-form wk on AI'"there corresponds a k-form h' (wk) on the open set V in the space R' orMm, respectively. If y = (yi, ... , y'") are the associated coordinates, thenh'(wk) can be represented by means of the component functions w/ as

h*(wk) = Ewldy1.I

The summation extends over all ordered multi-indices I = (il <... < ik),where dy1 is a shorthand notation for the k-form dy1 := dy`' A ... A dy'k.

Definition 11. A k-form wk on the manifold Mm is called a differentialk-form or smooth k-form if for each chart h : V - Mm the coefficients w1of the k-form h*(wk) are smooth functions on the subset V C IIt".

Differential forms can be added and multiplied by functions defined on A1min an obvious way. Hence the set S2k(Alm) of all C°°-forms of degree k is amodule over the ring C°O(AIm) of smooth functions on the manifold.


The exterior derivative of k-forms-familiar from the euclidean space ;,F"and discussed in Chapter 2-can now be transferred to the situation of k-forms on manifolds without difficulties, preserving all the known properties.This proceeds as follows: For a k-form wk on Al' and a chart h : V - 11"'we first consider the induced form h'(wk) and its differential d(h'(wk)). Thelatter is a (k+ 1)-form on the set V, and its pullback under the inverse chartmap h-1 : It (17) -- V is a (k + 1)-form. This yields a (k + 1)-form

dwk := (h-1)`(d(h*w'))

defined on the open set h(V) C 1M'". The construction just described is in-dependent of the particular choice of the chart, and hence it uniquely definesa global (k + 1)-form dwk on the manifold M. In fact, for another chartIt, : V1 -. Mm satisfying h(V) fl h1(V1) 0 0, we obtain on the intersectionh(V) n h1(V1) the equality

(hl 1)'(d(hiwk)) = (h-')*(hi 1 o h)'(d((h-1 o hl)`h'(wk)))

= (h-1)d((h, 1 o h)'(h-1 o h1)*h'(wA))

= (h-1)d(h*wk) .

This computation relies on the fact that the exterior derivative commuteswith the diffeomorphism (hi 1 o h), a property discussed in Chapter 2.

Definition 12. The (k + 1)-form dwk defined starting from the k-form wkon Al... is called the exterior derivative of wk.

All properties of the exterior derivative known from euclidean space remainvalid in the situation of a manifold. The next theorem summarizes them.

Theorem 13. For arbitrary forms wk, Wk' 17I on a manifold M.. the follow-

ing properties hold:

( 1 ) d (wk + wk) = dwk + dwi ;

(2) ddwk = 0;

(3) d(,,;k A tl') _ (&,k) A,/ + (-1)Awk A (d77');

(4) if f : N" -+ Mm is a smooth map, then f' commutes with theexterior derivative, d(f*wk) = f'(dwk)

The purely algebraic operation of forming the inner product between a vectorfield V and a k-form wk is also transferred into the situation that both objectsare defined over a manifold.

Definition 13. We define the inner product of a vector field V and a k-formwk by

(V _J wk)(W1, ...,Wk-1) := wk(V,W1.....Wk-1).

3.4. Orientable Manifolds 69

3.4. Orientable Manifolds

An orientation of a real vector space is the choice of one of the two equiva-lence classes in the set of its bases discussed in the first chapter. This can, inparticular, be applied to the tangent space T=M' of a manifold, and leadsto the notion of an orientation 0= at the point x E Alm. An orientation ofa manifold Mm consists in a "continuous" choice of orientations at each ofits points.

Definition 14. An orientation 0 of a manifold Alm is a family 0 = {Or} oforientations in all tangent spaces TM m depending continuously on the pointx in the following sense: At each point x E Alm there exists a chart h : VMm containing this point such that the basis {h. (a/ay' ), ... , h. (a/aym) }is compatible with the orientation Oh(y) for every point y E V.

Definition 15. A manifold Al' is called orientable if there exists at. leastone orientation on it.

First we state a necessary and at the same time also sufficient condition forthe orientability of a manifold in terms of the chart transition maps.

Theorem 14. Let 0 be an orientation on Mm. Then there exists a family{(hi, V )liEI of charts with the following properties:

(1) The image sets hi (Vi) cover the manifold,

film = U hi(V) ;iEI

(2) If the intersection hi (V) n h; (V3) 34 0 is non-empty, then the de-terminant of the differential of the chart transition map h, 1 o hi ispositive,

det[D(hj 1 o hi)] > 0.

Conversely, if there exists a family of charts with these properties, then Mmis orientable.

Proof. Let 0 be an orientation on Mm. Choose those of the charts h :V Alm for which the basis {h.(a/ayl ), ... , h .(a/ay`)} is compatiblewith the orientation of the manifold. By the definition of an orientationthese charts cover the manifold. For two of these charts, hi : V Afm andh3 : Vj -+ Mm, with coordinates y = (y1, ...Iym) and z = (z'...., zm) wehave at mutually corresponding points

{hi.(a/ayl), ... , hi.(a/ay')} = Oh;(y) = Oh,(z)

_ {hj.(a/azl),...,hl.(a/azm)}.

70 3. Vector Analysis oil Manifolds

Therefore, these bases are compatibly oriented, the transition matrix be-tween them is the differential D(h, 1 o h;), and we obtain

det[D(hj 1 o h;)] > 0.

The converse is proved analogously.

If the manifold All C IR" is described by (n - m) independent equations,then it is orientable. We will prove this fact now.

Theorem 15. Let f1, .... f"_,,, : U , R be smooth functions defined on anopen subset U C 1R". and assume that

1df1 A ... A dfn-in T 0at each point. Then the manifold

Mm = {rEU: fl(x)_...=fn-n(x)=0lis orientable.

Proof. Consider the euclidean gradients grad(fl ), ... , grad(fn_,,,) of thefunctions. By assumption, these vector fields are linearly independent ateach point of the set U and, moreover, perpendicular to the tangent spaceTjlll"' at the points of the manifold (compare Theorem 5). An orientationOx in TiAfm is distinguished by requiring that a basis v1, ... , vm E TAf"'is positively oriented if and only if

d]R"(grad(fl)(x) ....grad(fn-,.)(r),v1, ...,v, ) > 0.Here dR = dx1 A ... A dx" denotes the volume form on R". It is not hardto see that this condition determines all orientation on M"'.

An oriented submanifold Al'" c R" carries a distinguished differential formof highest degree, the so-called volume form. Choose a basis e1.... , em ETxM"' consisting of mutually perpendicular vectors of length one in thefixed orientation Ox at any tangent space TIM . For arbitrary vectorsill i .. , v,,, E Tx:'ll 'n we define

(vi,el),dill'"(v1....,vm) = det

(v,, el),This definition uniquely determines the form dAl". Every other basisel, in the same orientation can be represented as a linear combi-nation e; = F_ A,2ej with an orthogonal matrix A of positive determinant.But then det(A) = 1.

Definition 16. The m-form dMm on the oriented manifold Mm C R" iscalled the volume form.


Remark. The volume form dAlm is not the exterior derivative of an (m-1)-form. Nevertheless, this form is traditionally denoted by dMm. The volumeform does not vanish at any point of the manifold. Evaluating dMm on anyorthogonal basis e1, ... , em in the orientation yields

dMm(e1, ..,em) = 1.

On the other hand, the orientation can be reconstructed from the volumeform. In fact, a basis v1, ... , vm is positively oriented if

dt11m(vl, ...,vm) > 0.

Changing the orientation of the manifold results in a change of sign for thevolume form.

Theorem 16. An m-dimensional manifold Mm is orientable if it carries anowhere vanishing differential form of degree in.

Proof. For an orientable manifold Alm the volume form dAlm has the nec-essary property. Conversely, suppose that there is an m-form Wm on AI"' notvanishing at any point. Then we call a basis v1, ... , vm positively orientedif

Wm(vl, ...,v",) > 0.This determines an orientation on Al m. 0

In a chart h the induced volume form h*(dM'") is proportional to dy' A... A dy"', h' (dMm) = f (y)dy1 A ... A dym. We compute the function f (y)as follows:

f2(y) = (h`(dMm)(a/ayl, ... , a/aym))2 = det2 [(h.(a/ay'), e,)]

= det [(h.(a/ay'),ej)J det [(ej,h.(a/ay'))]= det [(h.(a/ay'), h.(.9/ay'))] = det[g;l]

Hence f2 (y) is equal to the determinant det[g;j (y)] of the Riemannian metric,and we obtain the formula

h*(dM-) = 9(y)dy' A...Adym.Example 24. For a surface piece A12 C R" with a parametrization h][1;2 , AI2 C R", the following classical notation is frequently used:

= / ah ah \

G = 922(X, Y)

E = 911(x,y) ate, Y-ah ah

( -.9Y 09-Y

ah ahF = 912(X, y) = a-, 5


The formula for the volume form is then

dM2 = EG-F'2dxAdy.

Example 25. The volume form of 1R' in cartesian coordinates is

(14) d1W = dx1 A... Adx".

Example 26. In polar coordinates on 1R2 - {0} we have g(r, gyp) = r2, andhence the volume form is

(15) dIR2 = r . dr n dip .

Example 27. In spherical coordinates on the 2-sphere, 1P) = cost v/i,and thus the volume form is

(16) dS2 = cos t dip A diP.

In the case of an oriented manifold, the divergence of a vector field V canbe expressed in terms of the exterior derivative and the volume form.

Theorem 17. Let Mm C R" be an oriented manifold, let dMm be its volumeform, and let V be a vector field with divergence div(V). Then

d(V J dMm) = div(V) dM'".

Proof. In local coordinates we start fromm

V = Vs(y)aii and dMm = /dyl A...Adyt.

i=1 y

This impliesm

V-i dMm = (-1)'-1V'dy' n...AdyiA...dy'",,Fg

i=1

and the formula

d(V.j dMm) = ma(/

V t) dy' A ... A dym = div(V) dMmi=1

immediately follows from the definition of the divergence of a vector field.0

Remark. As in the euclidean space R. for every (oriented) manifold M'"the 1-form w , dual to a vector field V can be described using the volumeform and the Hodge operator:

*wv := V i dM' .

The divergence formula can then be written as d(*wl,) = div(V) dM'".


Non-orientable manifolds exist; proving non-orientability for a particularmanifold, however, is sometimes a little more difficult than to show ori-entability. We state a simple criterion for the non-orientability of a manifold.This can be used, e.g., to prove that the Nlobius strip is non-orientable.

Theorem 18. Suppose that. for a manifold Mm, there exist two chartsh : V Mm and h : V M' with the following properties:

(1) V and f7 are connected sets, and the intersection h(V) n h(V) isnot connected;

(2) the determinant det(D(h-' o h)) has opposite sign at two points inh-'(h(V) n h(V)).

Then M' is a non-orientable manifold.

V

Proof. Assume that MI is orientable. The sets V and V are open subsets of]R'", and hence orientable. Thus, without loss of generality, we may assumethat the chart transition maps h and h preserve the orientations. Considerthe volume form dMt, and represent it in both coordinate systems:

h* (dM-) = J dy' A ... A dyt , h' (dM-) = f dz' A ... A dz' .

Note that in both these cases the sign is "+", since the chart transitionmaps h, h preserve the orientation. Because

(h-1 o h)'(dz' A ... Adz') = det(D(h-' o h)) dz' A ... A dzm,

we havef det(D(h-'oh)) = f.

But this is a contradiction, since f and Vg' are always positive. 0

The orientation of a manifold induces a unique orientation on its boundary.To define it, we make use of the exterior normal vector field. Its definitionrelies on the following observation.

Lemma 2. Let U C ]H' be an open subset of the half-space. let x E U n 8HHmbe a point in the boundary OHHt, and let f : U - V be a dieèomorphism fromU to an open subset V C H. F o r a vector v = (x, (v', ... , v' )) E T=H"'


at the point x with non-positive m-th component, v"' _< 0, the image vectorMV) = (f (x), (w1, ..., w"')) also has non-positive m-th component w"' < 0.

f. (v)

Proof. Choose a straight line -y(t) = x + t v and note that, because ofv"' < 0. the point y(t) belongs to U fl HH"' for sufficiently small negativevalues of the parameter t. Then we have y(0) = x, y(0) = v. and the in-thcomponent of f.(v) becomes negative:

f"'(x+tv)-0_dtf"'(x+tv) = lim < 0. 0t-o- t

At a boundary point x E 8M' C M' of a manifold there are two tangentspaces. On the one hand, there is the tangent space TxMm of Al'. and,on the other hand, the boundary determines its own (in - 1)-dimensionaltangent space, T=(8M'") C TXM'". At each boundary point x E 8M'. wedefine a unique tangent vector N(x) E TxMm by the following conditions:

(1) N(x) is perpendicular to T

(2) N(x) has length 1;

(3) for a chart h : V C Him - Mm around the point x E Al"' the m-thcomponent of the vector (h)-1(N(x)) is negative.

Definition 17. The vector field N constructed along the boundary is calledthe (exterior) unit normal vector field of the boundary.

d


We fix an orientation of the boundary OMm of an oriented manifold M' bycalling a basis vl, .... vm_1 E TT(OMm) positively oriented if the extendedbasis A((x), vi, is positively oriented in T2A1m. A simple argumentshows that this condition actually determines an orientation of the boundary.For the corresponding volume forms we have the important formula:

-

Theorem 19. The volume form of the boundary of a manifold is the innerproduct of the exterior normal vector field N with dull'":

d(aMm) = Ni dM'n

Proof. Choose an orthonormal basis in the orientation of the vector spaceT,,111 consisting of the normal vector el := N(x) together with additionalvectors e2, ... , e,n. For arbitrary tangent vectors v1.... , 0m-I to the bound-ary we obtain

(Ari ditlm)(t'1, ...,v1) = d111"'(el,vl, ...,vrn-1)(vi, e2) ... (vl,em)

= det(vm-1, e2) ... (vm-1, em)

= d(OMm)(i'1, .. , l'm-1) ,

since the vectors vl, ...,v,,,_1 are perpendicular to el. 0

In the subsequent sections of this book the boundary of a manifold willalways be oriented this way.

Example 28. The exterior unit normal vector field of the sphere Sn-I(R)of radius R, considered as the boundary of the ball Dn(R), has the form

1r(la nalN(x)

= R(x ax-1+...+x

ax's

The volume form of Sn-I(R) can also be computed by means of dSn-1(R) _N.J dR', yielding the formula

1(17) dSi-l(R) = E(-1)'-Ix'dxlA...A dx'A ...Adx".i=1

Hence the volume form of the sphere S"-'(R) can be represented via theembedding i : Sr-1(R) -+ IR" as the form induced from the following (n-1)-form defined on Rn:

,,n-1 = -1x'dxl A ... Adx' A ... Adxn'i=1

i.e., the following equation holds:dS"-1(R) = i.(wn-l)


Computing the exterior derivative of wi-1 in R' leads to the formula

dwn-1 = n dlRn ,R

and by applying Stokes' theorem from Chapter 2 we obtain the integral ofthe volume form:

J dSn-1(R) = J dwn-1 = Rvol(D°(R)) .n-l(R) D"(R)

3.5. Integration of Differential Forms over Manifolds

The integral of an m-form over an m-dimensional manifold will be definedby dividing the manifold into small subsets which are diffeomorphic to opensubsets of R' or Hn', respectively, and integrating the given form using itschart representatives one by one. The sum of the resulting values is then thetotal integral of the form over the manifold. From the very beginning we willconfine ourselves to compact manifolds in order to have to deal with finitelymany summands only and thus to avoid convergence questions for seriesarising otherwise. A detailed exposition of this definition of the integralrequires a so-called partition of unity in order not to count contributionsfrom overlapping charts twice. This is a special family of smooth functions tobe discussed first. Recall that the support of a function cp : Mn' - Ris the closure of the set {x E Mm : V(x) 710).

Theorem 20. Let Mm be a compact manifold. Then there exist smoothfunctions and charts

cpi : A f '- 1R and hi : V - Mm (1 < i < l)

with the following properties:

(1) The support of the function vi is contained in hi (V ),

supp(wi) C hi(V)

(2) The functions Wi are non-negative, and their sum is equal to one:

Ws(x) 1 .

i=1

Proof. We choose a chart h., : Vr - M"' around each point x E M nsatisfying h,,(0) = x. Here, the Vi are open subsets of Rn or H'n. Choose,moreover, non-negative functions 1G:: R'n, H'n R such that

(1) z(0) = 1;

(2) supp(r!.?) C V.

3.5. Integration of Differential Forms over Manifolds 77

Setting0 if y V hx(VV) ,

;Px(y) =1Gx(hx 1(y)) if y E hx(VV),

these functions can be transferred to form a family of non-negative smoothfunctions bx : M' --+ R on M. By construction we have c3x(x) = 1 andsupp(,px) C hx(VV). Taking advantage of the compactness of Mm, we finallyobtain finitely many functions 31, ... , yet and charts (hl, V1). ... , (h1, VI)satisfying

tAr = I {xEMm:yi(x)j4 01

ii=11

and supp(cpi) C hi(Vi). The sum i(, < is positive at each point, andthe functions cpi we are looking for result from normalizing these functions'A' Vi = cpi/v.

Now we will define the integral of an m-form w' over an oriented andcompact manifold M. To this end, we choose charts hl, .. . , ht and apartition of unity, Cpl, .....pt subordinate to them satisfying the propertiesformulated in the preceding theorem. Furthermore, we suppose that thechart maps hi : Vi -+ M'" preserve orientation. Then

fi(y)'dylA...Adymis an rn-form on V with compact support, and we eventually define theintegral:

t t

Jwm jhWm) _ Jf(y).dy1...dym.

it=1 t=1

We show that this definition of the integral is independent of the chosenpartition of unity. Consider another atlas (h1, V1), ..., (h,., V,.) with subor-dinate partition of unity 'p1, ... , ;p,.. If the intersection of the chart rangesis not empty, hi(V) f1 ha(V0) 0, then

fi(0pWm) = jhQ.Jm).This follows immediately from Theorem 7 in Chapter 2, since the deter-minant of the differential D(h=1 o ha) is positive. Summing now over theindex i, 1 < i < 1, as well as the index a, 1 < a < r, we obtain fromE Bpi =_ E cpa = 1 the relationi a

gym) _ f hi (vi ' gym)a=1 V i=1 V,


Definition 18. The m-dimensional volume of a compact and oriented sub-manifold Mm C lR" is the number

fvol(Mm) := J dM'".

M^'

Consider a curve, viewed as a 1-dimensional submanifold M1 C IR" of R".If It : (a. b) - M1 is a parametrization of the curve, then

bjdM'a

= Zand we recover the length of the curve. More generally, if It : V - If'is a parametrization of Mm, then, using the coefficients of the Riemannianmetric, the volume of M' can be written as

vol(M'n) = J det[g,.1 (y)] dy.V

Example 29. The last formula in §3.4 shows that the (n - 1)-dimensionalvolume of the sphere Sn-'(R) and the n-dimensional volume of the ballD"(R), both of radius R, are related by

vol(S"-1(R)) = Rvol(D"(R)).

Example 30. The coefficients of the Riemannian metric on the torus ofrevolution discussed in Example 4 are

(ri + r2 cosV)2, 9vv = 0, g = r2 .

Hence f = r2(rl + r2 cos V), and we obtain21r 2ff

vol(T2) = fJr2(ri + r2 cos cp)dcpdi' = 47,2r1 r2 .

0 0

We will conclude this section by a remark concerning measure theory onmanifolds. Let CO(Mm) be the ring of all continuous functions defined onthe compact and oriented manifold Mm. Setting

µ(f) :=JMm

f dM-

defines a linear functional µ : CO(Mm) - lR which is positive and monoton-ously continuous in the sense of the following properties:

(1) If f E C0(lbfm) is a non-negative function, then µ(f) > 0.(2) If f,, is a monotonously increasing sequence converging to a function

f E CO(Mm), then

nlim

o µ(f") = µ(f)-.a

3.6. Stokes' Theorem for Manifolds 79

This turns the pair (CO(Mm), p) into a so-called Daniell-Stone functionalon the set M. Within the framework of general measure theory, one firstassociates with every, D-S functional p on a set X an outer measure p' onX. The Caratheodory construction then leads to a a-algebra of subsets inX on which p` becomes a measure. By applying these general principlesto a compact and oriented manifold, one constructs the so-called Lebesguemeasure on MI. It is defined on a a-algebra containing the Borel sets, andevery bounded and Borel-measurable function turns out to be integrable.For the purposes of vector analysis we do not need this extension of thenotions of measure and integral to manifolds, since the functions occurringare, as a rule, at least continuous. It is, however, interesting to note thatthe notion of integral treated here fits into the general theory of measureand integration as sketched above. The interested reader may refer to theliterature for details (see the book of K. Maurin in the bibliography).

3.6. Stokes' Theorem for Manifolds

In the preceding sections we collected all the notions necessary for the for-mulation of Stokes' theorem on manifolds.

Theorem 21 (Stokes' Theorem). Let Mk be a compact, oriented manifold,and suppose that the boundary OMk is endowed with the induced orientation.Then, for every (k - 1) -form wk-1 on Mk,

f wk-1 = f dwk-1

8Mk JAlk

If, in particular, Mk has no boundary, then for every (k - 1) form wk-1

AIkdwk-1 = 0.

Proof. In Theorem 20, it was shown that we can choose finitely many charts(V1, h1), ... , (Vr, hr) covering the manifold 11Ik, together with a subordinatepartition of unity O 1, ... , cpr : Mk R. We label these charts in such away that, on the one hand, for each index i less than a certain index ro theset V intersects the boundary, V n 8Mk 34 0, and, on the other hand, forall indices i > ro the set V is disjoint from the boundary, V n OMk = 0.From >i vi = 1 we obtain >i dcpi = 0 and use this to rewrite the exteriorderivative of wk-1:

k-1=

k-1 Pidw- k-1 + dipi n wk-1

i=1 i=1 i=1 i=1


This implies the equationr

JM' dwk-1 = d (hi (Piwk-1))

V is an open subset of Rk for rp + 1 < i < r, and h; is a k-form onV1 whose support is completely contained in V. For each of these indices iwe choose a k-chain c in Rk for which

supp h, (Wiwk-1) C Int ck C ck C V j.

Applying now Stokes' theorem for chains (Theorem 8, Chapter 2) to ck, weobtain

fd (ht

wA-1))

=Jd(h(iwk_1)) = J h' (Viwk-1) = 0,

1. ask

since the form vanishes on the boundary of the chain. Now we consider theindices i between 1 and ro. For any of these, V is an open subset of the half-space Hk, and as before we obtain chains ck in lHlk with the same propertiesof the supports with respect to h; Applying Stokes' theorem tothese as well, we obtain

1 d (hiJ

hi (ca1wk-1) .

v;

Now hi (,p1wk-1) does not necessarily vanish any more on Ock n (Rk- l x {0} ),

but only at the points of 8c; belonging to the interior of Hk:

d (h; (Vjwk-1)) _ hi (`Piwk-l).

VflRk-1V; IThe pairs (V n Rk-1, h1IRk) with i = 1, ..., ro form a covering of the bound-ary BAIk. Hence, by the definition of the integral,

Jamk

ro

wk-1knRl-- t-1 1

h i (ViwR-1) ,

showing the equation we set out to prove. O

In the sections to follow we will be dealing with various applications ofStokes' theorem. As a generalization of the discussion in §2.5, we will firststudy line integrals and prove an analogue-only for 1-forms, however-ofPoincare's lemma. This holds for manifolds in which every closed path canbe contracted to a point.

Definition 19. A connected manifold Mk is called simply connected if everytwo C'-curves co, cl : [0, 1] , Mk with coinciding initial and end points arehomotopic.

3.7. The Hedgehog Theorem (Hairy Sphere Theorem) 81

By Theorem 9 in §2.5, whose proof immediately carries over to the case of amanifold, on a simply connected manifold the line integral of a closed 1-formw1 depends exclusively on the end points of the curve. Having fixed a pointxo E Mk, the line integral along a curve joining the points xo and x,

wf(x) =lox

uniquely defines a function on the manifold. Its differential df coincides withw1, and we obtain

Theorem 22. Every closed 1-form on a simply connected manifold is exact.

Example 31. The winding form defined on 1R2 - {0} is closed, but notexact. This shows that R2 - {0} is not simply connected.

3.7. The Hedgehog Theorem (Hairy Sphere Theorem)

Consider two oriented compact manifolds Mk and Nk without boundaryand of equal dimension. Two maps fo, fl : Mk - Nk between them arecalled homotopic if there exists a smooth map

F : Mk x [0,1] -+ Nk such that F(x, 0) = fo(x) and F(x,1) = fl (x).We prove

Theorem 23. Let wk be a k-form on Nk and let fo, fl : Mk Nk behomotopic maps. Then

JMk0 (wk) = fm kfl (wk) .

Proof. The oriented manifold Mk x [0, 1] has boundary

8(Mk x [0,1]) = Mk x {1} - Mk x (0},

where the minus sign indicates that the orientation is reversed. Therefore,Stokes' theorem implies

J kfl (w) -

JMk 8f(wk) =

J(Mkx[O,1])F*(wk) =

JMk x [OI]dF(wk) .

f

But the form dF*(wk) = F'(dwk) = 0 vanishes, since the k-dimensionalmanifold Nk carries no non-trivial (k + 1)-forms. 0Theorem 24. The antipodal map from the sphere to itself, A : Sn - Sn,A(x) = -x, is homotopic to the identity Ids. only for odd dimensions n.

Proof. Consider on Rn+1 the formn+1 _

wn = -1x' dx1 A ... A dx' A ... A dx"+1i=1


whose restriction to the sphere Sn is its volume form dSn (Example 28,equation (17)). If A is homotopic to the identity Ids.., then the previoustheorem implies

J A*(wn) = / n w" = vol(Sn) .

is. S

The induced form A* (,n) = (-1)n+lwn is proportional to the form n.Thus we obtain the condition

(_1)n+lvol(Sn) = vol(S"),i. e., (n + 1) has to be an odd number.

Theorem 25 (Hedgehog Theorem). A sphere S2k of even dimension hasno nowhere vanishing, continuous tangent vector field.

Proof. Suppose that there exists such a vector field on the n-dimensionalsphere S". We first approximate this vector field by a smooth vector fieldV (Stone-Weierstrass theorem), and then normalize it so that the vectorV(x) has length one at each point. Next we consider the resulting smoothtangent vector field as a vector-valued function V : Sn - Rn+I satisfyingthe following two conditions:

(x, V(x)) = 0, IIV(x)II = 1.Define the homotopy F : Sn x [0, 1] - Sn from the sphere to itself by theformula

F(x, t) = cos(irt) x + sin(irt) . V(x).The length of F(x, t) is equal to one everywhere, since x and V(x) are per-pendicular. Moreover, F(x, 0) = x and F(x, 1) = -x, i. e., F is a homotopybetween the identity and the antipodal map of the sphere Sn. But then thedimension n has to be an odd number.

In the German mathematical literature, this result is known as the "Hedge-hog Theorem" (,,Satz vom Igel"), since its contents can be expressed figu-ratively by saying that a hedgehog cannot be combed in a continuous way.Because it is so vivid, we prefer this to the name "Hairy Sphere Theorem",which seems to be more common in the Anglo-Saxon world.

3.8. The Classical Integral Formulas

Now we will discuss the classical integral formulas already treated in §2.6 forchains. Compared to the preceding case, in this new formulation we benefitfrom having notions like divergence, gradient and Laplacian as developedin the differential calculus on manifolds at our disposal. At the same time,the orientation of the manifold and the exterior unit normal vector field onthe boundary will play a special role. We will prove these integral formulas

3.8. The Classical Integral Formulas 83

for arbitrary compact and oriented manifolds (in R'). This will be the finalformulation of the classical integral formulas as they are needed in manybranches of mathematics as well as theoretical electrodynamics and hydro-dynamics. We start with the Ostrogradski formula relating the divergenceof a vector field to its flow across the surface.

Theorem 26 (Ostrogradski Formula). Let Mk be an oriented, compactmanifold, and let N be the exterior unit normal vector field to its boundary.Then, for every vector field V : Mk Tlllk on Mk,

div(V)dMk = (V, N) d(OMk).

lL nik

Proof. We know from Theorem 17 that the divergence and its inner productwith the volume form are related by the formula

div(V) dMk = d(V J dMk) .

A straightforward application of Stokes' theorem implies

J div(V)dMk = J d(V i dMk) = f Vlk k nl k

Let x E 8Mk be a point of the boundary. We decompose the vector V(x)into one part that is proportional to the exterior normal vector, and a vectorW(x) that belongs to the tangent space T,ZBMk to the boundary:

V(x) = (V(x), N(x)) N(x) + W(x) .

Moreover, note that the restriction of the inner product W J dMk to theboundary 8Mk vanishes identically. This implies that for the inner productof V with the volume form dMk. on the boundary 8Mk

V _j dMk = (V, N) N j dMk + W i dMk = (V, Al) N(x) _j dMk .

Hence, we obtainJ

fdiv(V)dMk = (V, N) Ni dll'ik .

ik Mk

By Theorem 19, the inner product N(x) . dMk coincides with the volumeform of the boundary, d(8Jik). 0As a direct application of the Ostrogradski formula we obtain Gauss' theo-rem.

Theorem 27 (Gauss' Theorem). Let V be a vector field, let f be a functionon the oriented, compact manifold Mk, and let N be the exterior unit normalvector field of the boundary. Then

J l (V, grad(f )) dMk + f k f . div(V)dMk = Jalk f (V, N) d(81bik) .

l


Proof. This equation immediately follows from the Ostrogradski formulatogether with the rule from Theorem 11

div(f V) = f div(V) + V(f) = f div(V) + (V, grad(f)) .

In a similar way we derive Green's formulas in versions that are not confinedto 12. First we generalize Green's first formula.

Theorem 28 (Green's First Formula). Let f, g : Mk R be smooth func-tions on the compact, oriented manifold Mk. Then

f f'O(g) dMk + g'ad(g)) dMk = f f (grad(.9)N) d(8Mk).

A1k aMk

Proof. By the definition of the Laplacian, we have

J %rkf O(g) dMk = f'k f div(g'ad(g)) dAik .

Now apply Gauss' theorem.

Applying Green's first formula twice leads to Green's second formula.

Theorem 29 (Green's Second Formula). Let f, g : Mk - III be two smoothfunctions. Then

1 [g. (f)-f. (g)] dMk = f N) ] d(aMk).

Ark aMk

1-1

Remark. The scalar product (grad(f ), N) defined only on the boundary isoften denoted by the symbol Of ION, since it is the derivative of the functionf in the direction of the exterior normal vector. This leads to a differentformulation of Green's second formula:

f [g . (f) - f . %(g)] dMk =fMk

[g.-f.]iaN aN

Corollary 1. Let Mk be a compact, oriented manifold without boundary.Then

(1) f div(V) dMk = 0 for every vector field V;

Mk

(2) f go(f) dMk = f fo(g) dMk = - f (grad(f ). grad(g)) dMk

Alk Mk Mk

for any two functions f, g E CO°(Mk).

0

3.8. The Classical Integral Formulas 85

Hence the Laplacian is symmetric with respect to the L2-scalar product.Moreover, the choice of sign we adopted implies that it is non-positive.

Corollary 2 (Hopf's Theorem). Let Mk be a compact, connected, orientedmanifold without boundary and assume that the function f : Mk R sat-isfies at each point the condition 0(f)(x) > 0. Then the function f isconstant.

Proof. Integrating the assumption s(f)(x) > 0 over Mk and applying thesymmetry of the Laplacian just proved, we first obtain

0< f 1-o(f)-dMk = Jf-j(1)-dMk = 0,

f k Mk

i.e., the Laplacian of f vanishes identically, A(f) = 0. Inserting f = ginto Green's first formula (Theorem 28) and taking into account that theboundary integral vanishes by the assumption concerning Mk, this implies

J grad(f)I2-dMk = -Jntkf 0,

and hence grad(f) = 0. Thus f is constant, since Mk is connected. 0

Concluding this section, we formulate Stokes' theorem in its classical formon 1R3. Contrary to the preceding theorems involving the generalizations ofdivergence and gradient to manifolds as introduced in the second section ofthis chapter, this only involves the notion of curl on open subsets of 1R3 from§2.3.

Theorem 30 (Stokes' Theorem-Classical Version). Let M2 C R3 be acompact, oriented surface, let V be a vector field defined on an open subsetM2 C U C 1R3, let N : M2 S2 be the exterior unit normal field to thesurface M2, and let T : aM2 - T(aM2) be the unit tangent vector field onthe boundary curve aM2 with the induced orientation. Then

f (curl(V), N) dM2 = f MZ (V, T) d(0M2).

Proof. Consider the 1-form

4 := V1dx1 + V2dx2 + V3dx3

on U associated with the vector field V = V la/axl + V2a/axe + Via/ax3as explained in §2.3 and its derivative

d`4 =av2 aV1 , 2 aV3 avl , 3 aV3 aV2 2 3

axl - axe I dx ndx + ax, -ax3

] dx ndx + [ax2 - ax3

J dx ndx .


Recall that the curl of V corresponds to the 1-form *dwv. If, on the otherhand. It : W -+ M2 is a parametrization of the surface with componentsh', h2, h3 and coordinates yl, y2 from W, then for the exterior normal vectorN to the surface we have the relation

09

ll

h x 09h

/ II A x ahyl y2 09y1 y2

Two arbitrary vectors v, w E R3 satisfy the identity

111Iv x wII2 = det (V, V) (v, w)(v, w) (w, w) J

and hence the preceding equation implies

II aylx

aye IIf.

For the first component of the normal vector N written in the coordinatesy1, y2 this reads, e.g., as

N'dM2 = N' f dy' n dye = 1 ahe Oh3 _ah2 ah3 dye-Vrg, dy' A .

[ay' aye aye ayl

On the other hand, it is easy to compute the pullback by h of the forms dx2and dx3:

h*(dx2) =OhY12

+ A2dy2, h`(dx3) = ldy' + Oh3dy2.

A direct comparison implies the following formula, which is independent ofthe coordinates y', y2:

N'dM2 = dx2 A dx3 .

Similarly one proves

N2dM2 = - dx' A dx3, N3dM2 = dx' A dx2 .

The scalar product of the curl of V with the unit normal field N multipliedby the volume form dM2 is thus simply the differential of wv:

(curl(V), N) dM2aV3 aV2

1 2 aV' 9V3 2 2 aV2 aV' 3 2= [axe - 09x3

] N dM + 109x3 09x1, N dM + 09x1 - axe, N dltil= dwv. L L

Therefore, Stokes' theorem can be applied in the format

J (curl(V),N) dM2 = J&4 = y=J[V1dx1 + V2dx2 + V3dx3].

%f2 Af2 8M2 8M2

3.9. The Lie Derivative and the Interpretation of the Divergence 87

If, however, T is the unit tangent vector field to the curve 9M2, thend(8M2)(T) = 1, and hence

T'd(aM2) = dxl, T2d(8M2) = dx2, T3d(8M2) = dx3.

Now we can rewrite the line integral above as

J [V ldxl + V2dx2 +V3dX3]= J1L12

(V, T) d(8M2),

and, summarizing, we arrive at Stokes' classical integral formula. 0

3.9. The Lie Derivative and the Interpretation of theDivergence

The aim of this section is to interpret the divergence of a vector field geo-metrically as the infinitesimal volume distortion of its flow. First we recallsome results from the local theory of ordinary differential equations and in-troduce the flow on a manifold as well as the Lie derivative of forms. Thenwe compute this Lie derivative by means of the exterior derivative, which,in a special case, leads to the interpretation of the divergence mentioned inthe title.

Let V be a vector field on the manifold Mk. An integral curve of V is acurve ry : (a, b) Mk whose tangent vector -y(t) = ry,(8/8t) at each pointcoincides with the value of the vector field there:

?(t) = V('Y(t))

The well-known existence theorem for autonomous differential equationsstates that for every initial point x E Mk there exists a maximal integral


curve ryx : (ax, bx) - Mk defined on an interval, containing the number0 E R, satisfying

'Y.,(0) = X.Moreover, this maximal integral curve is uniquely determined by the initialcondition. Denote by EV the set

EV = {(t,x)E]RxMk:ax<t<bx}and call the map -0: EV Mk defined by the formula

,Dt(x) = '(t,x) := 7x(t)the flow of the vector field V . If the maximal integral curves are defined forall values t E R of the parameter (i.e., if Ev = R x Mk), then the vector fieldV is called complete. A complete vector field V determines a one-parametergroup 4Pt : Mk ---+ Mk of diffeomorphisms from the manifold to itself:

4)to+tl = 4t0 O 46tl .

This relation between diffeomorphisms is a direct consequence of the unique-ness of the integral curve for a fixed initial condition. Conversely, if a one-parameter group of difeomorphisms is given, then we obtain a vector fieldby looking at the tangents to the trajectories +t(7) of a point:

V (X) dt (bt(x))1t=o

Example 32. The one-parameter transformation group in R2 of the vectorfield

V = -x2871 +x1872

is determined by the differential equationsi1 = -x2 th2 = x1

and coincides with the group of all rotations in the plane.

Example 33. The one-parameter transformation group of the vector field

V= x1871

+...+x"ban

in ]R" is determined by the differential equations

.i1 = x1, ...,in = 7n

and coincides with the group of all dilatations in Ilt".

Suppose that the manifold Mk is a closed subset of R. Then Mk is a com-plete metric space: Every Cauchy sequence in Mk converges. For boundedvector fields we will prove


Theorem 31. Every bounded vector field on a complete manifold withoutboundary is a complete vector field. In particular, every vector field on acompact manifold without boundary is complete.

Proof. Denote the maximal length of the vector field V by K,

K := sup{IIV(x)JI: xEMk}.

Let : (a, b) , Mk be a maximal integral curve of the vector field. and as-sume that b < oc. We apply the mean value theorem of differential calculusto the vector-valued function y : (a, b) - AIk C IR". It states that for everytwo values of the parameter, a < t1 < t2 < b, there exists one more value,tl < t' < t2, such that

Ily(tl)-y(t2)II <_ It2-tll Ily(t*)II = It2-tll'IIV(y(t'))II <-Since b is finite, any sequence t; E (a, b) converging to b is a Cauchy sequencein R. The last estimate shows that the image sequence -y(ti) E Mk is also aCauchy sequence. By assumption, Mk is a complete manifold, and hence thesequence -y(ti) converges in Mk. This observation shows that the left-handlimit,

lim y(t) _: x`,tbexists. The point x` E Mk is an inner point of the manifold, since there areno boundary points by assumption. We choose a chart h : Rk -+ Mk aroundthe point x' = h(O) and represent the vector field V in the correspondingcoordinates,

V = Vla/eyl + ... + Vka/Oyk .

The solution y(t) of the differential equations

y1 = V 1(yl, .. . yk), .. , yk = V k(yl, ... , yk)

converges for t - b to the point 0 E Rk. Hence the integral curve y(t) canbe extended beyond the parameter b. This contradicts the maximality of bresulting from the assumption b < oc. 0

Now we turn to the Lie derivative of a differential form with respect to avector field. If w' is an i-form, then. pulling it back by the flow of the vectorfield, we obtain a 1-parameter family of forms, 4 (w'). The derivative of thatwith respect to the parameter at t = 0 is the so-called Lie derivative. Thisconcept involves the flow of the vector field only in a small neighborhood ofzero and is thus well-defined for arbitrary vector fields.

Definition 20. The Lie derivative CV (w') of an i-form w' is the form

CV(w`) li owt


Theorem 32. The Lie derivative of a differential form is expressed in termsof the exterior derivative and the inner product as

'CV (d) = d(VJ w')+VJ (dw`)

Proof. The exterior product w' A 171 is bilinear, and the product rule isproved as usual,

Cv(w' Ar?) = Cv(w') AT? +W' ACv(T?)

Furthermore, the exterior derivative d commutes with the induced maps 4)i,and hence we have

Cv(dw') = d(Cv(w))Denote the right-hand side of the equation to be proved by C4(J). Then

Cv(dw') = d(V J dw') + V J (ddwl) = d(V J dw')= d(VJdw'+d(VJw')) = d(C*(w')).

The first, purely algebraic equation,

VJ (w'Ar7') _ (VJ w')Ar7+(-1)'w'A(VJ r'),leads to the identity

Cv(w'Ar7') = d(VJ (w'Arl'))+VJ d(w'Ar1')= d((VJ w')Aq'+(-1)'w'A(VJ r?))+VJ (dw'Ar7'+(-1)'w'Adr7')= d(VJ w')Ar7'+(-1)'-'(VJw')Adr7'+(-1)'dw'A(VJ r))

+w'Ad(VJ+(-1)'(V J w') A dr7' + w' A (V J dri)

= (d(VJ w')+VJ r7')+VJ dr7')

= Lv(w')Arl'+w'ACv(r')Lastly. C and C coincide for functions,

Cv(f) = V(f) = VJ df = G`v(f),and thus the formula is proved. 0

We apply the formula of the preceding theorem to the volume form of anoriented manifold Alk. From the definition of the divergence of a vector fieldwe obtain

Cv(dMk) = d(V J dAlk) = div(V) dAfk.Fixing a point x E Afk and choosing a neighborhood U(x, e) C Mk of radiusEJiv(V).dM'> 0, we obtain by integration

1= I l=

(l,e) /


Dividing both sides by the volume of the set U(x, e) and taking the limit forE - 0. we see that the mean value theorem of integral calculus immediatelyimplies the formula

div(V)(x) = limvol(U(x, r)) (vol(t(U(x.e)))) .

u=o

This formula allows a geometric interpretation of the divergence of a vectorfield: It is the volume distortion of its flow in infinitesimally small time andon infinitesimally small domains surrounding the point x.

Corollary 3. The divergence of a vector field vanishes if and only if its flow4t : AIk _ Mk consists of volume-preserving diffeomorphisms.

Proof. For a measurable set A C Afk we denote by hA(t) := vol((Dt(A)) thevolume of the set -(Dt(A). The derivative of the volume change is computedas follows:

d (hA(t 61t =dtd

I f dAfk) =L1A)d(A) =dt

f t (4)Gv(dMk) = 41 div(V) dAlk.

(A)

From this we see that the flow 4bt preserves the volume if and only if thedivergence of the vector field V vanishes identically. 0

The Lie derivative of a vector field W with respect to another vector field Vis defined just like the Lie derivative for forms; nevertheless, it is called thecommutator of both vector fields.

Definition 21. Let V and W be two vector fields, and denote by theflow of the vector field V. The commutator [V, WI is defined as the vectorfield

[V, W1 = limW

t-0 t

We prove an important formula relating the commutator of two vector fieldsto the exterior derivative of 1-forms.

Theorem 33. Let wl be a 1-form, and let V, W be two vector fields. Then

dwl(V,W) = V(wl(W)) - W(w'(V)) -w'([V, WI)

Here V(wl(W)) is the derivative of the function w1(W) in the direction ofthe vector field V.


Proof. If x E Mk is a fixed point, then by the definition of the commutator[V, W]

[V, W](x) = Iiox))) - W(x)

.

The formula to be proved then follows from Theorem 32 by means of thefollowing computation:

wl([V W])(x) = lim'

w'(W(x))t--o t

l= lio

(4'-t)'(f') - w'(W(4'c(x))) + u ow'

(W(t(x)) - W(x))

_ -(Cvw')(W)(x) + V(w'(W))(x)_ -W(w'(V))(x) - dw'(V,W)(x)+V(w'(W))(x). 0

If, in particular, wl is a closed form, then we obtain

w'([V, W]) = V(w'(W)) - W(w'(V))In local coordinates the vector fields are represented by their components,

kV=viaApplying the above formula to the closed form dyt gives us

k

dy'([V,W]) _V;aWl - k W1.

i=1 i==11

This leads to the following local expression for the commutator of two vectorfields:

[V W] _,

' 'ay ay1J \ I1-Example 34. The commutator of the vector fields V = y a/ax, W = 0/4defined on 1R2 is the vector field [V, W] = -8/0x.

The algebraic properties of the commutator of vector fields are summarizedin the following theorem.

Theorem 34. Let U, V, W be vector fields, and let f be a smooth functionon the manifold Mk. Then the following identities hold:

(1) [U + V, W] _ [U, W] + [V, W];

(2) [V, W] _ -[W, V];

(3) [U, [V, W]] + [V, [W.U]] + [W, [U, VJ] = 0 (Jacobi identity);

(4) [f V,W] =f

k (Viewt - Wi avt) l a


Proof. These formulas immediately result from the local expression for thecommutator. We present the calculation for the last of the formulas:

k vitaw,a f-VI)l a[f v, w] _ E (i.v'ayt -

w ayt J'-l_ (F _) ( k -vi awl wi av a wt of (vi) af ayi ayi ) ay,

ay, EOV0=1 i=1 1=1

=

In the following sense, the commutator is a natural operation between vectorfields.

Theorem 35. For a diffeomorphism 4' : Mk -+ Mk and vector fields V, Won the manifold Mk the differential (P. is compatible with the commutator:

'F, ([V, W]) _

Proof. Note first that for every 1-form w' the functions 4'(w')(W) andw' ('. (W)) are related by the formula

'DO(wl)(w) = w'(=(w)) 04-' -

Theorem 33 then implies

wl ((I).[V+w]) = ('F*wl) ([V, W]) O'D

= V ((''wl)(W)) o 4)- W o 4)- d(4)`wl)(V, W) o 4)

_ ('F.V) (w'(4'.W)) - ('.w) (U" (4%v)) - (dw') ('.v, 4.W)= w' (['.(V),4'.(W)])

This equation holds for every 1-form w', and hence the theorem follows.

Remark. Requiring that 'F be a diffeomorphism is often too strong. Thecompatibility of the commutator with the differential can sometimes beproved under weaker assumptions. For example, if the map 4) : M - N issmooth, the vector field V on M is said to be -t-related to the vector fieldW on N if at each point m E M

$.,m (V(m)) = W('F(m)).

If one can now show that, together with the pairs (VI, WI) and (V2, W2),the commutator pair ([V1, V2], [WI, W2]) is necessarily also fi-related, thesame compatibility property follows immediately. To see this, apply the


commutator [WI, W2] to a function g : N --+ R:

[WI, W2]$(m)(9) = WI(4i(m))(W29) - W2 (4)(m)) (WI 9)

= 4'.,m(VI(m))(W29) - 4)4,m(V2(m))(WI9)

= VI (M) ((W2 9) 0 4') - V2 (M) ((WI 9) 0 4?).

On the other hand, (W1 g) o 4? = Vi(g o 4?), since

V+(n)(9 o 41) = 41.,n(Vi(n))(9) = Wi(4)(n))(9) = (Wi 9)(4'(n))

Thus we obtain

[W1, W24(m)(9) = [Vl, V2]m(9 0 4>) = 4).,m[Vl, V2]m(9) ,

i.e., the vector fields [WI, W2] and [VI, V2] are, in fact, 4)-related.

The commutator of two vector fields measures the extent to which theirflows do or do not commute. This explains the name for the vector field[V, W].

Theorem 36. Let V and W be two complete vector fields on the manifoldMk and denote by 4't and X8, respectively, their flows. Then the commutator[V, W] vanishes if and only if 4it o 4' = %P, o tt for all -oo < s, t < oo.

Proof. Because

d= tim (4'-h-h).W - (4'-t1) /

= (4'-t1).([V, W]),

the commutator [V, W] vanishes if and only if the vector field W is invariantunder the flow 4it, (4)1).W = W. This condition is in turn equivalent to thecommutativity 4't o T, = T. o 4it of the diffeomorphism 4't with the flow ID,of W.

3.10. Harmonic Functions

A function f : Mk - R is called harmonic if it is a solution of the homo-geneous Laplace equation 0(f) = 0. As a special case of Hopf's theorem(Corollary 2), we have the following:

Theorem 37. Every harmonic function on a compact, connected, and ori-ented manifold without boundary is constant.

If the boundary of the manifold Mk is not empty, there exist two particu-larly important boundary value problems for harmonic functions.

3.10. Harmonic Functions 05

The Dirichlet Problem: Assume that a function V : OMk - IR is given. Welook for a harmonic function f : Mk IR whose values on the boundaryaMk coincide with those of cp:

0(f) = 0 in Mk and f IaaMk = cp .

The Neumann Problem: Assume that a function cp : OMk -+ IR is given. Welook for a harmonic function f : Aik R whose normal derivative on theboundary coincides with yp:

0(f) = 0 in Mk and = cp on aMk.eiv

A solution of the Neumann problem is never unique. For each solution fthe sum f + C is, for an arbitrary constant C, a solution of this problem,too. This is the only degree of freedom, since we have

Theorem 38. Let Mk be a compact, connected, and oriented manifold inR", and let cp : aMk -+ IR be a smooth function.

(1) The Dirichlet problem has at most one solution.

(2) If f,, f2 both are solutions of the Neumann problem, then f, - f2is constant.

(3) The vanishing of the mean value of cp is a necessary condition forthe solvability of the Neumann problem:

JOMkp d(aMk) = 0 .

Proof. If fl, f2 both are solutions of the Dirichlet problem, then the differ-ence u := f, - f2 satisfies the equations

0(u) = 0 and u IBMk = 0.

From the first Green formula we obtain

0 = J -Lk U. a" d(aMk),Nfk 85Jk aNand hence the gradient of u vanishes. Since we have u IBMk= 0, the func-tion u has to vanish identically. In the case of the Neumann problem theargument runs along the same lines. i.e., starting from the equations

0(u) = 0 andau = 0 on aMk

and the Green formula, we again conclude that grad(u) = 0. Thus thedifference u = fl - f2 is constant. If the Neumann problem has at least one


solution for any given function cp :8Mk R, then, applying the classicalintegral formulas, we obtain

.d(OMk).0 f O(f) dMA = r LMk . d(Wk) = J

160

The Dirichlet and the Neumann problems have, for a given boundary condi-tion cp : 8Mk R (satisfying f cp = 0 in the case of the Neumann problem)a solution. We will not prove this existence result here for general mani-folds, but confine ourselves to the case of the ball D"(R) C R' of radiusR > 0. For these spaces, there is an explicit classical solution formula whichwill be derived here. For the sake of simplicity we only discuss the case ofdimensions n > 3 and leave it to the reader to complete the discussion indimension n = 2, which differs only slightly from the one below. We startwith a few preparations. By

r = (x1)2 + ... + (xn)2

we denote the distance from the point (x', ... , x") E Rn to the point 0 E Rn.

Lemma 3. Let u(x) be a harmonic function defined on the set Rn - {0},and assume that u depends only on the radius r. Then there exist constantsC1 and C2 such that

u(x) = C1 +C2

-n-2

Proof. By assumption there exists a smooth function h : (0, oo) -' R satis-fying u(x) = h(r(x)). Differentiating this, we obtain the formulas

Ou x' OZU n( )(' 112 i( )r2 - (xi)2

8xh(r)r, ax = h r \r/ +h r r3

and the latter implies the following differential equation for the functionh(r):

n _0 O(u)

52x1= lh(r).

For n > 3 the function h(r) = C1 + C2 r2-n is the solution of this ordinarydifferential equation.

Let y E Rn be a fixed point. The function u(x) = JJx - y112-n defined onthe set Rn - {y} is a translate of r2-n, and hence a harmonic function. Onthe other hand, the partial derivatives of a harmonic function are harmonicfunctions as well. Thus

u(x) - 2au I Ix112 I lyl l2

.=1 ax'y

Ilx yl In


is a harmonic function defined on iRn - {y}. We will use this family ofharmonic functions in the observations to follow.

Lemma 4. Let IIxii < 1 be any point in the interior of the unit ball D".Then

I n 1 IIxIIXII" dSn-1(y) = vol(Sn-').

Proof. First we prove that the function2h(x) .

_f n-' IIx IIyIIn dSni-1(J)

depends only on the radius r = r(x). In fact, for a linear orthogonal map T :1R" -+ R" we obtain, from I det(DT)I = I det(T) I = 1 and the correspondingtransformation formula for the volume form of the sphere,

T*(dSn-1) = T*(NJ d1Rn) = NJ (T'(d1Rn)) = NJ dRn = dSn-1,

the property to be proved:

h(Tx) = Jn-1 IITxIITyIIn

dSn-1(y) = J n-s lixlTII'(Iy)Iin dSn-1(y)

_ I 1- IIXII2 . dSn-1(z) = h(x).Jsn-' IIx - zlln

If, in addition, h(x) is a harmonic function, then

fz (h)(x)= -

A. (1_ilx2yIIn

dSn-1(y)) = 0.IS,1

The first lemma implies that there are constants C1, C2 with h(x) = C1 +C2 r2-". At the point x = 0 the function h(x) is a regular function andsatisfies h(O) = vol(Sn-1). Thus the constant C2 vanishes, and C1 is equalto vol(Sn-').

The solution of the Dirichlet problem for the unit ball D' C ]R" togetherwith an explicit formula are the subjects of the next theorem.

Theorem 39. For every continuous function cp : S"-1 - 1R, the formula

1 12AX) =

_

vol(Sn-1) is., IIx

II-

yII" cp(y) dS"-'(y)

defines a harmonic function in the interior of the unit ball, and at a boundarypoint z E Sn-1

lim f (t z) = cp(z).t-1-


Proof. The function (1 - IIxII2)IIx - yll-,, is harmonic with respect to thevariable z, and hence the function f (x) is harmonic, too. We prove thatf(z) coincides with ;p(z) on the boundary Sn-1 in the way stated above.Denote by in := y E Sn-1} the maximum of the modulus of pand fix a positive number e > 0. The continuous function p is uniformlycontinuous on the compact set Sn-'. Hence there exists a number b > 0such that for any two points y, z E Sn-1 in the sphere, i i - z < 6 impliesthe estimate I :p(y) - cp(z) I < e. We decompose the sphere Sn-1 = D1 u D2into the parts

D1 = {y E S" : IIy - ziI < b}, D2 = {y E Sn-1: II - zII > a} .

For y E D2 and 0 < t < 1 we estimate the distance from y to the linesegment between 0 and z E S"-':

fly - tz1I2 =

> 1 - (y, z) = 2-(2- 2(y, z)) = -Ily - zII2 >2

.

1 - 2t(y,z) + t2 = (t - (y,z))2 + 1 - (y,z)2 > 1 - (y.Z)2

1 1 b2

We then split the difference2_

P ' z) - Y (z) = vol(Sn-1) J$^ IltzY dSr (?/)

into the integrals over D1 and D2. The modulus of the first integral can beincreased to

1-IItzII21 n 1dS"

Di S^-' Iltz - yllnWe treat the second integral using the inequality obtained before:

< 2m - (1-t2)(a2)n

vol(Sn-').D2

In summary, for every positive number E > 0 there exists a number 6 > 0such that for all 0 < t < 1 the following inequality holds:

E+2mb - t2).

Hence the upper limit is bounded by E,

lim suplf(t' z) - V(z)I < E

The last inequality holds for all positive numbers E > 0, and this in turnimplies

lim f (t z) = p(z) . 0tt-


Next we will discuss several consequences of the solution formula for theDirichlet problem. For a harmonic function f : D"(R) IR defined on theclosed ball of radius R > 0, the function j (z) := f (R z) is also harmonicon the unit ball. Applying the previous theorem and returning to the vari-able r = R z E D"(R) finally leads to the Poisson formula for harmonicfunctions:

2 2 - 2

f(x) = vo1Sn-1) J "-' IIR- RI yl l1"f(R y) dS"-1(y)

Evaluating the Poisson formula at the point x = 0 leads to Gauss' meanvalue theorem for harmonic functions.

Theorem 40. The value of a harmonic function f at the center of the ballcoincides with the mean value of the harmonic function on the boundary ofthe ball:

We will use Gauss' mean value theorem in the proof of the maximum prin-ciple for harmonic functions.

Theorem 41. Every harmonic function f on the connected manifold Mn CR" attaining its maximal value in the interior of Mn is constant.

Proof. Denote by m the maximal value of f and by Q = {x E Al"\8AI" :f (x) = m} the set of all inner points of M" at which f attains this maximalvalue. By assumption 1 is a non-empty closed set in M"\811". Hence itsuffices to prove that 1 is an open subset of M"\8M". Choose a pointxo E S2 and a radius Ro such that the ball D"(xo, R0) with center x0 andradius Ra is completely contained in Mn\8M". By Gauss' mean valuetheorem

f (x0) =VOl(S1

n-1) Jsn_l f (x0 + Ro y) dSn-1(y) m. = .f (xo)

This implies that f is constant and equal to m on the sphere with centerx0 and radius Ra. This observation can also be applied to each radiusRR < Ro below R0. Together this implies that f - m is constant on the ballD"(x0i R0), i. e., D" (x0, Ra) is contained in Q. Thus Il is an open subset. 0

Eventually we will prove one more application of the Poisson formula, Li-ouville's theorem for harmonic functions.

Theorem 42. Every harmonic function f : 1[P" R bounded from below(above) is constant.


Proof. Changing f, if necessary, by adding a constant, we can supposewithout loss of generality that the function f is non-negative. Fix a pointxo E Rn and choose the radius R such that xo lies in the ball D"(0, R). Bythe Poisson formula, Gauss' mean value theorem, and the assumption f > 0we have

Rn 2 R' - II 2

f (xo) = vol(Sn-1) Jsn I I o - R yll In . f (R y) dS"-' (y)

<

Rn-2

J R2 - I IxOI I2n .f (R y) dSn- ' (y)

vol(Sn-1) n-i IIIxoII - IIR ylll

_ 2( 2 - II 2

IsnvoSn-RIIlxolloIIRIn f(R y) dSi-1(y)

RSn

-2(R2 _

)IIIxoII

- iRI" f(0)

Taking the limit for R -* oo yields for all points xoi E R" the estimate

f(xo) <- f(0),and the maximum principle for harmonic functions implies that f is con-stant.

3.11. The Laplacian on Differential Forms

This section is a supplement to the vector analysis on manifolds as explainedbefore. We will discuss the Laplacian acting on forms and its properties.To this end, we will start from an oriented manifold Mm of dimension mtogether with a family of non-degenerate scalar products

g=:TXM' xTIMRand assume that these scalar products depend smoothly on the point. Thisis understood to mean that for any two smooth vector fields V and W onM"' the function g(V, W) is smooth. Such a family {gx)xEAfm is called apseudo-Riemannian metric on M. In general, it has to be neither positivedefinite nor the restriction of the scalar product on the ambient euclideanspace.

In this situation the tangent space is a real oriented vector space. and wedenote by k the index of the scalar product. According to Chapter 1, at eachpoint there exist a volume form dMm(x) E A ,'(M') and a Hodge operator

'* : A (Mm) - Ax'-'(Mm)Together we obtain an m-form dMm on the manifold, as well as a *-operatorassociating with every i-form a smooth (m - i)-form. The algebraic rules

3.11. The Laplacian on Differential Forms 101

from Chapter 1 can still be applied. We will use these and Stokes' formulato determine the operator b : S i+1(Mm) -+ 1'l (Mm) adjoint to the exteriorderivative d : S1'(MTh) - S1'11 (M') acting on i-forms. As is well-known,the adjoint operator is characterized by the requirement

J m(w',17'+1) . dMm = ! (wi bi7i+1) . dMm .

f)fm

Here wi and rli+1 are forms with compact support disjoint from the boundaryof M', and (dwi, gi+1) denotes the scalar product induced on forms. ByTheorem 5 in Chapter 1 we have

(dwi dMm = (-1)k&' A (*1]i+1)= (-1)kd(wi A (*,ni+1)) + (_1)k+i+lwi A (d * 17i+1)

Integrating this equation and once again applying the algebraic rules, weobtain

fm-(dwi,,gi+') , dIim = (_I)k+i+1(-1)i(m-i)+k

Jwi A * * (d * 37i+l)

Mm

= (_1)$(m-i)+i+1+k (wi *d * qi+1) , dMm.nMm

Thus we have proved

Theorem 43. The operator 6 adjoint to the exterior derivative d : S2i(Mm) --,11i+1(Mm) on a pseudo-Riemannian manifold of index k is given by

b(lli+1) = (-1)k+mi+l * d * ni+1

Definition 22. Let Mm be a pseudo-Riemanniani manifold. The Hodge-Laplace operator acting on i-forms, A: S2i(M') 1'(M'), is defined by

°db+bd.Example 35. For a function wO = f, in the sense of this definition,

°f = bdf = (-1)k+l *d*df,and, in the case of a positive definite scalar product, ° coincides with theLaplace-Beltrami operator (see the remark following Theorem 11, §3.2).Considering, on the other hand, the pseudo-euclidean scalar product of sig-nature (n, 1) in Rn+1, we obtain for the Hodge-Laplace operator the expres-sion

i 2f f a2f°f = (92x1 +''' + a2xn a2xn+1

Rn+1 with this pseudo-Riemannian metric is called Minkowski space. Hence,its Laplacian on functions is the wave operator (also known as the d'Alem-bertian).


Definition 23. An i-form w' defined on a pseudo-Riemannian manifold iscalled harmonic if 0(w') = 0; it is called co-closed if bw' = 0.

In the following theorem we collect the properties of the Hodge-Laplaceoperator immediately resulting from the definition.

Theorem 44.

(1) Let w', if be forms with compact support disjoint from the boundaryof the manifold. Then

rl') dMm = r (w',0(rl')) dJV1-Im- (A (u;'),

(2) Aod=doA and Dob=boo.

Finally, consider a compact manifold without boundary endowed with aRiemannian metric (k = 0). Then the second order differential equation0(w') = 0 is equivalent to two first order differential equations.

Theorem 45. Let MI be a compact, oriented manifold without boundary,endowed with a positive definite metric. Then the following conditions areequivalent:

(1) w' is a harmonic form, A(w') = 0;

(2) w' is a closed and co-closed form, dw' = 0 = bw'.

Proof. If w' is harmonic, then

Jf

0 = rJ M'" f'^ 5fr

This implies dw' = 0 = bw', since the metric is positive definite. 0Corollary 4. Let Mm be a compact, oriented manifold without boundary,endowed with a positive definite Riemannian metric. If every closed 1 -formon M' is exact, then there exist no non-trivial harmonic forms.

Proof. In particular, a harmonic 1-form wl is closed, and hence wl can berepresented as the differential of a function, wl = df. Since

0 = bwl = bdf = 0(f) ,

f is a harmonic function, and by Hopf's theorem (Corollary 2) it is constanton each connected component of M. Therefore, its differential vanishes,w'=df=0. 0Example 36. The assumption of this corollary is, for example, satisfied formanifolds Mm which can be represented as the union M' = U U V of twoopen sets U, V C M'", where

3.11. The Laplacian on Differential Forms 103

(1) the intersection U fl V is connected,

(2) U and V are diffeomorphic to star-shaped domains in R"'.

In fact, Poincare's lemma implies the existence of two functions ft, : U - Rand fi : V R such that wl = dfu over U and wl = dfv over V. On theintersection we then have

d(fu - fv) = 0,and hence fuu - fi, = C is constant. Now

f(x) _ I fl(x)+C ifxE V.fu (x) if x E U.

is a uniquely defined function on AP" satisfying wl = df. For n > 2 thesphere S' allows a decomposition of the required shape, and hence S" hasno harmonic 1-form with respect to any Riemannian metric.

Example 37. Consider the 2-dimensional torus in 1R'1.

T2 = {(xi, s2, x3. xi) E 1R4 : (x1)2 + (x2)2 = 1, (x3)2 + (x4)2 = 1 }

and denote by i : T2 R4 its embedding into 1R4. The 1-forms induced onT2

wl := i'(-x2dx1 + xldx2) and n1 := i*(-x4dx3 + x3dxa)

are harmonic. To prove this, we use the parametrization h : [0,27r] x[0.2a[ -+ T2 defined by the formula

h(,jp, z') = (cos p. sin yp. cos i;'', sin r') .

In this chart,h`(wl) = dp and h'(gl) = dv',

and hence wl and 171 are closed forms on T2. Computing the coefficients ofthe Riemannian metric leads to the matrix

9iv 0

01 '

From this, it is easy to see that wi and ill correspond to each other via theHodge operator of the torus T2:

But then, wl and 91 are also co-closed, bwi = *d * wl = *dij' = 0.

Harmonic forms play an essential role in topology and global differential ge-ometry. Here we formulate the main result of Hodge theory; a proof wouldgo beyond the scope of this elementary text. For a compact, oriented Rie-mannian manifold without boundary, (Mm. g), we introduce the followingvector spaces:


(1) the space 7-l'(Mr,g) of all harmonic i-forms,

x' (M', g) = {w' : 0(w') = 0} ;(2) the space E(Mr) of exact differential forms,

E'(M') = {w' : there exists an (i -1)-form n`-1 such that d7ji-1 = w'} ;

(3) the space CE'(Mm, g) of co-exact differential forms

CE' (M'", g) = {w' : there exists an (i + 1)-form tl'+1 such that 67i'+1 = w' }

Theorem 46 (Hodge's Theorem).(1) The space lf(Mm,g) is a finite-dimensional vector space. It is

isomorphic to the de Rham cohomology HDR(Mm). For a fixedmetric, each cohomology class of closed forms contains preciselyone harmonic form.

(2) With respect to the L2-scalar product, the spaces W, E' and CE' areorthogonal subspaces of the space Q'(M) of all square-integrablei-forms and decompose it into the sum

fl' = W ®E' ®CE'.

From the Hodge decomposition we immediately obtain Helmholtz's theo-rem for 3-dimensional compact manifolds without boundary. Geometricallyspeaking, it states that each vector field can be represented as the sum ofa gradient field, a curl field, and a harmonic field. In particular, the vectorfield is the sum of a curl-free and a divergence-free vector field. In the finalchapter we will discuss and prove an analogous result for vector fields onR3 which decrease sufficiently rapidly at infinity within the framework ofelectrodynamics in detail.

Corollary 5. For each vector field V on an oriented, compact, 3-dimensionalmanifold M3 without boundary, there exist a function f , a vector field Wand a harmonic 1 -form ill such that

wsd(1) + wcurl(W) + 171

In particular, wy is the sum of a closed and a co-closed 1 -form.

Proof. Applying Hodge's theorem to the 1-form associated with the vectorfield V, we conclude that there exist a function f, a 2-form (32 and a harmonic1-form 771, together satisfying the equation

wv = df +0132+R1.

From §2.3, we know that the gradient of a function corresponds to thedifferential 4f,

1df = wsd(I)

Exercises 105

Let W be the vector field corresponding to the 1-form *i3`,2 1=: WW

The curl of the vector field W is the vector field determined by the equationdu) l 1

W = *wcurl(w)But this implies

a2 I 1 1

= *d * = * W = * * Wcurl(W) = wcurl(W)since in dimension m = 3 for a positive definite Riemannian metric therelation ** = Id holds on the space of 1-forms. This completes the proof. 0

Exercises

1. a) The equation

(x1)2 (x n2 +...+

2= 1, where al, ...,a, > 0 are constants,

a1 an

defines an (n - 1)-dimensional submanifold of R', the ellipsoid.b) The equation x2 + y2 = r2 (r > 0 fixed) defines a two-dimensional

submanifold of R3, the cylinder.

c) Consider for every c E R the subset of R3 defined by

M, = {(x,y,z)ER3: x2+y2-z2=c}.For which values of the parameter c is Al, a two-dimensional submanifoldof 1R3?

2. Prove or disprove the converse of Theorem 1: Let U C 1Rn be open, letf : U - Rn-k be smooth, and denote by Al the zero-set of f, Al := {x EU : f (x) = 0}. If the estimate rk D f (x) < n - k holds for at least one pointx E M, then M is not a submanifold.

3. Prove that the formula

x2 + y2 = a2 cosh 2(z/a), a > 0 constant

defines a two-dimensional submanifold of R3, which is, moreover, parame-trized by

h(v',v2) = (acosh(vl/a)cosv2. acosh(v'/a)sinv2, vl)This surface is called the catenoid in differential geometry.


4. Let f : C C be a complex polynomial f = Ek0 aiz` without doublezeroes. Consider for every natural number I > 2 the set

M = {(z,w)EC2: w'- f(z) = 0).Prove that Al is a two-dimensional submanifold of C2 = ]R4. The assignment(z, w) w uniquely defines a map G : M C satisfying G'(z, w) = f (z).The set M is a so-called Riemann surface. The l-th root of f (z) is uniquelydefined on it.

5. Consider the special linear group SL(2, R) of all real-valued (2 x 2) ma-trices of determinant 1:

SL(2,1R) = JA E M2(IR) : det A= 1 } .

Prove that SL(2, R) is a three-dimensional manifold, and that its tangentspace at the neutral element E is determined by

TESL(2,R) = {AEM2(R): trA=0}.

Hint: Apply Theorem 4 and the (well-known) formula det(exp A) = etr.a

6. Prove that the orthogonal group O(n,1R) consisiting of all orthogonal(n x n) matrices

O(n, R) = {A E A A' = E }

is a submanifold of 1Rn2, and compute its dimension. Is the subgroupSO(n.IR) of all orthogonal matrices of determinant I also a submanifold?If it is, compute its dimension, and describe its relation to O(n, R).

7. The set of all unit tangent vectors T1S2 to the sphere S2,

T,S2 = {(x, v) E IR3 X 1R3 : I = IIvII =1, (x, v) = 0 } .

is a submanifold of R3 x 1R3 which is diffeomorphic to the manifold

{(z'., z3) E C3: Iz1I2+ Iz2I2 + 1Z312 = 1, (z')2 + (z2)2 + (z3)2 = 0 } .

8. The product of a manifold M without boundary with an arbitrary mani-fold N again is a manifold, and it has the boundary 8 (M x N) = Al x 8N.

9. The tangent bundle TM of every manifold M is always an orientablemanifold.

Exercises 107

10. The topological boundary Fr(A) of a subset A C R" is the intersectionof the closure of A with the closure of its complement,

Fr(A) := A n R' \A .

Prove:

a) If M' C iR' is a closed m-dimensional submanifold of Rm, then OM' _Fr(M"').

b) If MI C 1R" (m < n) is a submanifold with or without boundary ofstrictly smaller dimension, then always Fr(Mm) = Mm.

11. Find three vector fields of length one on the sphere S3 which are mu-tually othogonal at each point.

12. Construct a vector field of length one on each sphere of odd dimension.

13. Compute the representation of the following vector field in polar coor-dinates:

V (X, y) = x2 + y2 (8/ax + 49/ay) .

14. Consider on the sphere S2 the vector field

V (X' y, z) = ((x, y, z), (-y, x, 0))

Prove that V cannot be the gradient field of any function f : S2 R. Onwhich more general fact is this example based?

15. Compute the local coefficients (gij) of the Riemannian metric as well asthe volume form of the following submanifolds in the respective charts:

a) for the sphere S" in the chart determined by the stereographic projectionh : 1R" -> S":

(2 yi 2 y" 1y12 - 1h(y', ..., y")

Jyi2 + 1'... +

JyJ2 + 1' 1y12 + 1) ,

b) for the pseudosphere, i.e., the surface of revolution defined by the trac-trix in the parametrization

h(u',u2) = (asinu' cosu2, asinu'sinu2, a(cosu' +lntan(u'/2)))

16. Compute the formulas for the gradient of a function, for the divergenceof a vector field, and for the Laplacian on the sphere S2 in the coordinatesdefined by stereographic projection.

108 3. Vector Analysis on manifolds

17. On an open subset of R3 (to be determined!) we introduce coordinatesby the chart

(rcosVcosv,rsin;pcos ,rsin;p)

on the parameter domain r E (0, oo), E [0, 27r], , E (-t'/2, o/2). Com-pute the coefficients of the Riemannian metric, and the formulas for thegradient, the divergence, and the Laplacian in these coordinates.

18. Let 111' and N' be k-dimensional manifolds. The map f : 111' , N'is called angle-preserving or conformal if, for two arbitrary vectors w1, w2 ETZM' and their image vectors vi = (W;) E TjiyiNA, the respective anglescoincide:

(y1, V2) _ (wl, w2)

]Iv1 11 -1111211 11w111. IIw211

If, in particular, (h, U) is a chart of the manifold Alk, the chart h is angle-preserving if and only if the representation of the Riemannian metric in thecorresponding coordinates is a multiple of the k-dimensional unit matrix E:

(gtj (y)) = A(y) . E, ,\(y) > 0.

19. Prove the orientability of the sphere S", using stereographic projectionand Theorem 14.

20. Let AI' be a manifold, h : U _ M' a chart, and -y : [a, b] --. h(U) C 111'a curve in Afk completely contained in the image set of h. Represent thecurve 7 in the coordinates (h, U) as h-1 o ry(t) = (y1(t), ... , y'(t)). Provethat the length of the curve can be computed by means of the coefficientsof the Riemannian metric g1, in the chart (h, U) via the formula

b 1/2

l(7) = f 9ti(7(t))d dtt) d dtt)dl.

a

21. Let f : [a, b] , R+ be a positive function and let Ale = {(x. y, z) E R3 :y2 + z2 = f2(x)} be the corresponding surface of revolution in R3. Provethe volume formula

/bvol(M2) = 2n / f (x) 1 + (f'(x))2 dx.

a

22. Compute the following surface integrals:

a) for M2 = (x, y, z E R3) : x2+y2+:2 = a2. z > 01:ty -

Exercises 109

b) / (x2 + y2) 012. where M2 is the boundary of the subset of 1R3hr

described by the inequality x2 + y2 < z < 1.

23. Compute the following surface integrals:

a) f (xdyAdz+ydzAdx+zdxAdy);2(R)

b)Jt2

[(y - z)dy A dz + (z - x)dz A dx + (x - y)dx A dy], where M2 is then

boundary of the subset of R3 described by the inequalities x2 + y2 <z2,0<z<1.

24. Let the 1-form w1 on the Riemannian manifold Mk be given in the charth : V - Mk by wl = Ej f, dy'. Prove the following formula for the adjointoperator:

1k ab(w) = t ay, (119-9*3fi)

25. Let Mk be a compact, oriented manifold without boundary. A functionf : Mk -+ R is called an eigenfunction of the Laplacian with respect to theeigenvalue A E R, if 0 f= A f. Prove:

a) If fl and f2 are eigenfunctions with respect to different eigenvalues,1\1 A2, then

(fl, f2) L2 = JM

k fl(x)f2(x) dMk = 0.

b) Let S2 C R3 be the sphere of radius 1, AR3 the Laplacian on 1R3, Athe Laplacian on S2, f : 1R3 -+ R a smooth function, and set r =

x2 + y2 + z2. Then2(A`3

f)Is2 _ (arj +2Of

+°S2(fIS2)

c) Let P : IIt3 -y 1R be a homogeneous polynomial of degree m whichis harmonic, i.e. P satisfies OR'P(x) = 0. Prove that PIS2 is aneigenfunction of the Laplacian on S2, and determine the correspondingeigenvalue.

26. Let Mk C 1R" be a compact, oriented manifold, and assume that fMk x [0, oo) -+ 1R is smooth. The heat equation is

ozf(x, t) = af(x, t)at


Prove that if f is a solution of the heat equation satisfying f (x, 0) = 0 forall x E Mk and f (y, t) = 0 for all boundary points y E 8Mk and all timest E [0, to], then f vanishes identically on the set Mk x [0, to].

27. Let Al"-1 be the boundary of a compact n-dimensional submanifold of1R". and N(x) the exterior unit normal vector field. If V E 1R" is a constantvector, then

f(N(x), V) dM"-1 = 0.

28 . Prove the following commutator relations for the Lie derivative:

[Cu, iw] = i(V,w] and [Gv, Lw] = G[v,wl .

Chapter 4

Pfaffian Systems

4.1. Geometric Distributions

For smooth functions f 1, ... , f,"_k : R' - R with linearly independentdifferentials, the equations

1Vrk....,c,,,-k = {T E R' : J1 l2) = C1, ..., f,,,-k(X) = Cm_k I

define a smooth k-dimensional manifold. Linearizing this, in general non-linear, system of equations by passing to the tangent bundle. we see thatthis manifold is described by the system

Tblc,.....c,,,-k = {v E T lR" : df1(v) = 0, ... , dfm-k(v) = 0 1.

These equations determine at each point of IR"' a k-dimensional subspaceof the tangent space to II2'. The resulting family of subspaces can also bedescribed by different systems of 1-forms w1.... , wm-k. For example, if (hj)is a matrix of functions with nowhere vanishing determinant, the 1-formsw; = E ik hj - dfj satisfy

TMk.....cn,_k = {v E TIRt : w1 (v) = 0, ..., 0}.

The level surfaces, however, cannot be recovered from the knowledge of theforms w, alone. In general, the problem arises under which conditions lin-early independent 1-forms wl, ... , wm-k describe a family of k-dimensionalmanifolds via the system of equations (Pfaffian system)

W1 = ... = wm-k = 0.

Frobenius' theorem provides a complete answer to this question. This chap-ter is devoted to its proof and the discussion of various applications. In

111

112 4. Pfafhan Systems

Chapter 5. we will apply it to prove the fundamental theorems of curve andsurface theory.

Definition 1. A k-dimensional geometric distribution (or Pfaffian system)on Al'" is a family Ek = {Ek(x)} consisting of k-dimensional subspacesEk(x) C Tj A1"' in the tangent spaces to Mm depending smoothly on thepoints in the following sense: For each point xo E M"'. there exist a neigh-borhood xo E U C Al' and vector fields VI, ..., Vk defined on U such thatEk(x) coincides with the linear hull of the vectors VI (x), .... Vk(x) at everypoint x E U.

Example 1. Every nowhere vanishing vector field V on AI' induces a one-dimensional distribution El(x) formed by all multiples of the vector V(x).Conversely, every one-dimensional distribution El is locally determined bya nowhere vanishing vector field.

Example 2. The linearly independent 1-forms wl, ... , w,n_k on Al' deter-mine a k-dimensional distribution by

Ek(x) = {vETAf' : wl(v)=...=W,,,_k(v)=0}.

In analogy to the integral curve for a vector field, we now introduce thenotion of an integral manifold for a distribution.

Definition 2. Let Ek be a k-dimensional distribution on M'T'. A k-dimen-sional submanifold Nk C Mm is called an integral manifold of Ek if thetangent spaces of Nk coincide with the spaces of the distribution:

T,XNk = Ek(x) for all x E Nk .

Definition 3. The k-dimensional distribution Ek is called integrable if itadmits at least one integral manifold through each point x E Al'.

Example 3. For a distribution Ek defined by linearly independent 1-formsW I , . . . , W,, -k as in Example 2, a submanifold i : Nk --+ Al"' is an integralmanifold of Ek if and only if the restrictions of the forms to Nk vanish,

i*(wl) = ... = i*(wm-k) = 0.

The local existence theorem for solutions of ordinary differential equationscan be formulated as follows:

Proposition 1. Every one-dimensional geometric distribution is integrable.

Example 4. Consider the nowhere vanishing 1-form w = x dy + dz on 1R3together with the 2-dimensional distribution determined by w,

E2 = {vETR3: w(v)=0}.We show that this distribution is not integrable.

4.1. Geometric Distributions 113

Proof. Suppose that E2 is integrable. Then there have to exist an openset W C 1R2 and a smooth map h : W R3 such that h'(W) = 0 andrank (D(h)) _- 2. For example, one can choose for h a chart of the integralmanifold. In the coordinates of R3, the map h = (h', h2, h3) consists ofthree functions, and the condition h`(W) = 0 is expressed on W C R2 by theequation

0 = h' dh2 + dh3 .Differentiating this expression and forming the exterior product with the1-form dh2, we obtain

0 = dh' A dh2, 0 = dh2 A dh3 .

We then multiply 0 = h'dh2 + dh3 once again by dh' and take into accountthe equation we already derived, dh' A dh.2 = 0, to arrive at

dh' A dh3 = 0.In summary, all twofold products vanish, dhi Adhj = 0, and this contradictsthe assumption that the differential D(h) of the map h : W -. R3 has themaximal rank two. Hence the 2-dimensional distribution in 1R3 defined byw = x dy + dz cannot have an integral manifold.

For higher-dimensional distributions on a manifold the problem thus arises:Under which conditions do they turn out to be integrable? The answer tothis question forms the content of Frobenius' theorem. In order to formulateit, we also need the notion of an involutive distribution.

Definition 4. A distribution Ek on the manifold Mm is called involutive if,for every pair of vector fields V, Won Mm whose values V(x), W(x) E Ek(x)at each point belong to the distribution, the commutator [V, W](x) E Ek(x)again has values in Ek.

Theorem 1 (Frobenius's Theorem). Let Ek be a k-dimensional distributionon the manifold Mm defined by the (m - k) linearly independent 1 -formsW1.... , Wm-k:

Ek = {VETMm: Wj(v)=...=Wm-k(V)=0}.

Then the following conditions are equivalent:

(1) Ek is integrable;

(2) Ek is involutive;

(3) for every point xo E Mm, there exist a neighborhood xo E U C Mmand 1 forms Oij defined on U such that

m-kdwi = ei j A W j

j=1

114 4. Pfaffian Systems

for1<i<m-k;(4) for all indices 1 < i < m-k, the following exterior products vanish:

dwi A (Wl A ... A Wm-k) = 0 -

Remark. The fourth condition occurring in Frobenius's theorem is calledthe integrability condition for the geometric distribution or the correspond-ing Pfaffian system.

Remark. A one-dimensional distribution is determined by m - 1 1-formsW1, .... and then dWi n (Wl A ... A Wm-1) is an (m + 1)-form on them-dimensional manifold. This has to be zero for trivial reasons, and theintegrability condition of Frobenius' theorem is automatically satisfied; seeProposition 1.

We first prove some of the simpler equivalences in Frobenius' theorem andproceed as follows: (2) (3) . (4) and the implication (1) (4).The next section will be concerned with the proof of the central assertion ofFrobenius' Theorem, i. e. the implication (3) (1).

Proof of the equivalence (3) (4). Suppose that there exist local 1-m-k

forms Oij such that dwi = E 9ij A wj. Thenj=1

m-kdWi A (WI A ... AWr_k) _ E Oij AWj A (WI A ... A Wm_k) = 0,

j=1

since the exterior square of any 1-form vanishes. Conversely, assume thatcondition (4) is satisfied. In a neighborhood U C MI of the point xo weextend the family of linearly independent 1-forms Wl, ... , Wm-k by adding1-forms r1 j, ..., r1k so that the combined family, {wl, , Wm-k, 111, . . . , nk ),forms a basis f o r A (Mm) at each point x of U. The 2-form dwi (1 < i <m - k) can thus be represented as

m-k m-k k k

dwi = E CQ0 WQ A Wp + E E Da j - Wa A nj + E Eji - rlj ^ nlQ,3=1 0=1 j=1 j,1=1

with functions CQo, DQj and Ejl. The condition dwi A (W1 A ... A Wm_k) = 0implies

k

E Ejl-njA171A(W1A...AWm_k) = 0,jd=1

and hence the coefficients vanish, Ejl = 0. If we introduce the 1-formsm-k k

3=1 j=1

4.1. Geometric Distributions 115

the exterior derivative dwi takes the desired shape,

m-kdwi = E Bia n wa .

a=111

Proof of the implication (3) (2). For any two vector fields V and Wwith values in the distribution £k, we have wi(V) = wi(W) = 0. Fromdwi(V. W) = V(wi(W)) - W(wi(V)) - wi([V. W]) we obtain

dwi(V,W) = -wi([V,W]).

If now condition (3) holds for the distribution £k, then

m-kdwi(V,W) _ > Oji A wj(V, W) = 0.

j=1

and hence all 1-forms w1, ... , w,,,_k vanish on the commutator [V, WJ. There-fore, this vector field takes values in £k, i.e.. the distribution £k is involu-tive.

Proof of the implication (2) ; (4). Let £k be a k-dimensional involu-tive distribution. The form dwi A (wl n ... Aw,,,_k) has degree (m - k + 2).Inserting (m - k + 2) vector fields into this form, we can assume, withoutloss of generality, that two of these vector fields have values in £k. Denotethem by V and W. Because of the involutivity of £k, we have

w;(V) = w;(W) = 0 and dwi(V,W) = -wi([V,W]) = 0.

Therefore, the exterior product dwi A (wl A ... A wm-k) vanishes on every(m - k + 2)-tuple of vector fields.

Proof of the implication (1) : (4). The proof of this implication pro-ceeds like the one before. Let £k be an integrable distribution, xo E Alla fixed point, and h : W -+ All a parametrization of an integral manifoldthrough this point. Then we have h*(wi) = 0 (1 < i < m - k), and thisimplies h`(dwi) = 0. Thus the 2-form dwi vanishes on the k-dimensionalsubspace £k(xo),

dwi IEk(zo)XEk(zo) = 0.Inserting again (m - k + 2) vectors into the form dwi A (wi A ... Awm_k), weconclude that at least two of these vectors-call them V and W-lie in thesubspace Ek(xo). Hence wj (V) = wj (W) = 0 and &&,i (V, W) = 0. As above,we conclude that the form dwi A (wl A ... A Wm_k) = 0 vanishes.


4.2. The Proof of Frobenius' Theorem

The implication (3) = (1), which is the core of Fobenius' theorem, is alocal statement. Hence, without loss of generality, we can suppose that themanifold Mm is an open subset of R. We will prove a slightly more generalresult from which the proof of this implication immediately follows.

Theorem 2. Let w1i ... , w,,,_k be linearly independent 1 forms on an opensubset M"' C Rm such that

m-kdwi = E B.ij A wj

j=1

for certain 1 forms Oij. Then there exist at each point x E Mm a neighbor-hood U C Mm of x and functions hj and fj defined on the set U satisfying

m-kw= =

j=1

Proof of the implication (3) (1). Let £k be a distribution with theproperty stated in condition (3). By Theorem 2, we can represent the formsw; in a neighborhood U of an arbitrary point zo E M' as

m-k

j=1

for certain functions. By assumption, the 1-forms w1, . . . , w,,,_k are linearlyindependent. Thus the differentials dfli ... , df,,,_k are linearly independentas well, and the set

Nk = (X E U : f1(x) = fi(xo), ..., fm-k(X) = fm-k(XO) }is a submanifold containing the point xo E M. At an arbitrary pointx E Nk, we determine the tangent space:

TxNk = {v E TM' : df1(v) dfm_k(v) = 0 }

C {v E TM' : w1(v) _ ... = w,,,-k(v) = 0) = Ek(x).

For dimensional reasons, the vector spaces coincide, i.e., Nk is an integralmanifold of the distribution £k through the point xo E Mm, and thus theintegrability of the distribution £k is proved. 0

Proof of Theorem 2. We proceed in two steps. First we reduce the proofto the case of a system of 1-forms w1, ... , wm_k in a special normal form.To this end, we represent the euclidean space R n as a product RI = Rk Xam-k and denote its points accordingly by y = (y', ... , yk) E Rk and z =

4.2. The Proof of Frobenius' Theorem 117

(z1, ... , zm-k) E Rm-k. It is sufficient to prove the claim in Theorem 2 for1-forms of the following special type:

k

wi = dz`-1: Aii(y,z)'dy' (1<i<rn-k).j=1

In fact, representing the linearly independent 1-forms wi in coordinates,in

wi = E Pia da ° ,a=1

we may first assume that the determinant of the part of the matrix (pij)containing the first (m - k) columns and rows does not vanish. Let (qjj) bethe inverse of this matrix, and consider the 1-forms

m-k m-k in in mkw{ _ gia'wa = giaPa3d23 = dr'- E d.. 3

-giapa3

0=1 a=1 3=1 3=m-k+1 0=1

Thus the forms w; are of the special type above. Moreover,m-k rn-k m-k m-k

dwi = E dgia A wa + qia . d41a = dgia A wa + Y, qia ' Ba 3 A W3a=1 a=1 a=1 a,3=1

m-k (M- k m-k

_ E E dgia + E gi03ae3ry A wa,a=1 3=1 3.y=1

and the system of forms wi, ... , wm-k also satisfies the condition

m-kdw; _ B a A UJ

a=1

for certain 1-forms O . If Theorem 2 now holds for this system of forms, weobtain functions h,j and fj* such that

m-kw;

=> h,j dfj .

j=1

Rewriting this leads tom-k

Wt = Pia ' wa =a=1

m-k m-k

E E 11aj ' Pia ' dfia=1 j=1

Summarizing, it is sufficient to prove Theorem 2 for forms of the typek

wi =j=1

118 4. Pfafan Systems

Without loss of generality, we may choose the origin, x = 0 in Rm, as thepoint x = (y, z) in whose neighborhood we want to represent the forms w;in this special way. For fixed parameters (y', ...,y") E Rk near 0 E Rk, weconsider the ordinary differential equation with initial conditions

k

(t) _ > A;j(t - y, z(t)) yt, z'(0) = z', 1 < i < m - k.j=1

Denote by F` (t, y, z) the solution of this differential equation that is uniquelydetermined near zero. These are m - k smooth functions in the variablest E R, y E 1Rk and z E ]R"'-k defined on a neighborhood of zero in R x RI.Moreover, for small values of the parameters we have F'(µ t, y, z) _F'(t, p y, z), since the functions

G'(t,y,z) F'(µ't, y, z)satisfy the initial conditions G'(0, y, z) = z', and they are also solutions ofthe differential equation corresponding to the vector p y,

k

Y, Z) = p'F"(p.t,y,z) _ A:,(lA't_

j=1

We introduce a coordinate transformation near zero in R- = ]Rk x ]R--k bymeans of the equations

u .= y, F(1,u,v) := z.The Jacobian of this transformation is

ay,

zIo((

LE 0la(u,v) 0 E J

since, by the differential equation and the initial condition,

az' I = OF' (1, O, v) I = 8v' =0v jo 8 .7 0

&0 0 .

This observation shows that both these equations determine a local diffeo-morphism from the space RI to itself near zero. We represent the 1-formsWI, ... , wm-k in the {u, v}-coordinates:

m-k k

w; _ Bi j (u, v) dut + j:Pij (u, v) - duj .

j=1 j=1Obviously it suffices to prove that the functions P;j - 0 vanish identically.First we derive the identity

k

0.j=1

4.2. The Proof of Frobenius' Theorem 119

To this end, consider the map 4i : R' x Rk(t, y) -p Rk x R' -k(u, v) from thespace R1 x Rk with coordinates (t, y) to the space Rk x R' -k with coordinates(u, v) defined for a fixed point v E R'"-k by the formula 40(t. y) := (ty, v).Then

k

V(wi) = E Pig (t y, v)(t dyi + y dt) .j=1

On the other hand, with respect to the (y, z)-coordinates on R, the map4i : R' x Rk(t, y) - Rk x R"'-k(y, z) is determined by

4'(t, y) = (t y, F(1, ty, v)) = (t y, F(t, y, v))

From the normal form,

k

wi = dz' - Aidj=1

91,

we obtain for the induced form V(wi) the new expression

OF' k OF' kV (w;) =

atdt+E-y

j=1 j=1

Comparing in 4' (wi) the coefficient at dt and taking the differential equationinto account leads to the identity

k k

0 =.i=1 i=1

In the final step of the proof of Frobenius' theorem, we now show that thefunctions Pij = 0 vanish. To do so, we again resort to the map 4'(t, y, v) :_(t y, v), but consider it this time as a map from R x Rk x RI-k 9 (t, y, v)to Rk x R°'-k 3 (u, v) (v is no longer assumed to be constant). The identityalready proved implies the formula

4)*(wi) =m-k k

j=1 j=1

We denote by Pi*j the function

P*j(t. Y, v) := t Pip(t y, u).

Thenk

4 (wi) _ E P'j(t, y, v) dyt + (terms in dv'},i=1

120 4. Pfafan Systems

and for the exterior derivative we obtain the expression

k 8Pd-P`(wi) _ at dt n dyj + {terms without dt}.

j=1

m-kBy assumption, there exist 1-forms satisfying dwi = F, 9,1 A wj, and we

j=1represent these as V (9ij) := H,jdt + {terms containing dy', dvj}. Thisleads to a homogeneous differential equation for the functions we are inter-ested in:

ps m-k

E H.PQ>Q=1

with the initial condition P,j(0, y, v) = 0. The only solution of this homoge-neous differential equation with the given initial condition is P;j (t, y, v) - 0,and this immediately implies Pij(t y, v) = 0. Thus the proof of Frobenius'theorem is completed. 0

4.3. Some Applications of Frobenius' Theorem

The simplest case is that of an (m-1)-dimensional distribution £'"-1 on anm-dimensional manifold Mm. If Eii-1 is defined by one nowhere vanishing1-form w, the integrability of the distribution reduces to the condition thatthe 3-form dw n w vanishes,

dwAw = 0.The method to explicitly integrate this (m - 1)-dimensional Pfaffian systemis based on looking for a so-called integrating factor and an application ofPoincare's lemma.

Definition 5. An integrating factor for the 1-form w is a nowhere vanishingfunction f : Mm IR such that the 1-form f w is closed,

d(f . w) = 0.Theorem 3. Let w be a nowhere vanishing 1 form on the manifold Mm.

(1) If there exists an integrating factor for w, then dw n w = 0. In thiscase the distribution Em-1 is integrable.

(2) If dw Aw = 0, then there exists an integrating factor for the 1-formw in a neighborhood of each point in M'".

(3) Locally, the integral manifolds of the distribution £'n-1 are the levelsurfaces of the function g determined from the integrating factor fvia the equations

d(f w) = 0, f -w = dg.

4.3. Some Applications of Frobenius' Theorem 121

Proof. The equation d(f w) = 0 implies df A w + f dw = 0. Multiplyingthis equation once again by the 1-form w leads to f dw A w = 0. Sincef # 0, we obtain dw A w = 0 as a necessary condition for the existence of anintegrating factor. If, on the other hand, dw A w = 0, then the existence ofan integrating factor follows immediately from Theorem 2, §4.2.

In dimension m = 2 the 3-form vanishes, dw A w = 0, for purely algebraicreasons. Thus

Corollary 1. Every nowhere vanishing 1-form on a 2-dimensional manifoldlocally has an integrating factor.

Example 5. Consider in R2 the differential equation

P(t, X) + Q(t, x) i = 0.

Near a point (to,xo) E 1122 at which P and Q do not vanish simultaneously,we have the 1-form

w =and its integrating factor f (t, x). The equivalent differential equation

(f P) (t, x) + (f Q) (t, x)± = 0is called the total differential equation, and the solution curves are implicitlydetermined by the equation

g(t, x) = const

with dg = f w. Frobenius' theorem now claims that it is always possibleto solve the original differential equation by using the outlined method. Itdoes not, however, provide an algorithm for finding the integrating factor.In simple cases this may be computed directly. If we can find, e. g., functionsF(t) and G(x) depending only on the variables t, x and satisfying

OP(t, x) OQ(t, x)Ox at

= Q(t, x)F(t) - P(t, x)G(x) ,-then f (t, x) = of F(t)dtef G(x)dx is an integrating factor.

Example 6. Consider the differential equation

(2t2 + 3tx - 4t)± + (3x - 2tx - 3x2) = 0.

We can choose F(t) = 2/t and G(x) = -5/x and obtain the integratingfactor f = t2x-5. The solutions of this differential equation computed usingthis integrating factor are the curves described by the equation

t3x-4 - 2 t4x-4 - t3x-3 = coast

(compare the figure on the next page).


-a .4 -0.2 0 0.4

The curves t3 - zt2 - tax = cx4 for different values of c

Assume now that on the manifold M'" we are given a Riemannian metric(, ) as well as an (m -1)-dimensional integrable distribution E'-1 described

by the 1-form w. Denote the vector field associated with the 1-form via theRiemannian metric by W. This is uniquely determined by either of the twoequivalent equations

*w = W j dM' or w(V) = (V, W) .

Using the vector field W, the distribution E'"-1 can be described asE"'_1 = {v E TM' : (v, W) = 0) ,

and hence W is orthogonal to each integral manifold Nin-1 of the distribu-tion. Normalizing the length of the vector field W to one, the volume formof each integral manifold is given by the formula

dN'"-1

= -W i dM'" .

We study the behavior of the integral manifold N'"-1 C Mm of the dis-tribution under the flow 4b:: M'" -i Mm of the vector field W. Computefirst the Lie derivative of the 1-form w with respect to the associated vectorfield W. Since w(W) = 11W112, the formula £w(w) = d(W J w) + W i (dw)implies

(Gw(w))(V) = W(w(V)) - w([W, V))


In particular, for a vector field V tangent. to the distribution this formulasimplifies to

(,Cw(w))(V) = - w([W, V1) (W, [W, V]) ,

and we arrive at

Theorem 4. The flow 4it of the vector field W maps an integral manifoldof the distribution E"`-1 to another integral manifold if and only if

V(IIW112)+dw(W,V) = -(W, [W,V]) = 0

for every vector field V on M with values in £'"-1

Corollary 2. Let the distribution Em-1 be defined by the closed 1 -form w.Then the flow of the dual vector field W transforms integral manifolds intointegral manifolds if the length I I W I

Iis constant on every connected integral

manifold.

Corollary 3. If the distribution E'"-1 is defined by a 1 form w of constantlength. and if, moreover, the flow of the dual vector field W transformsintegral manifolds into integral manifolds. then dw = 0. In this case 6'-1

locally consists of level surfaces of a function whose gradient has constantlength.

Proof. First, Theorem 4 implies W . dw = 0. At a point x E Alm, wechoose an orthogonal basis e1, ... , e,,, in the tangent space so that W isproportional to el . W = a el. Denoting by al, ... . o. the dual basis, theform w = a of is proportional to al. Represent the 2-form

du; = Eb;j o1Aat<J

in this basis. Since 0 = dw A w, the 2-form dw only contains the in - 1summands dw = b12 Q1 A (72 +... + bi 2al A a,,,. But then the condition

0 = W-i dw = a(b12a2+...+bjmam)

means that w is a closed form.

We compute the infinitesimal volume change of a compact integral manifoldN'"' 1 under the flow 4)t of the vector field W. The Lie derivative of thevolume form dN1-1 is

,Cw(dA"-1) = £w(III .WJdAm) = WJd(IIWII.WJdMm)

_ (div(W) - 2 W(ln IIWII2)) dN'"-1

and hence we obtain


Theorem 5. The derivative of the volume change of a compact integralmanifold N` of the distribution Em-' under the flow of the vector fieldW is given by the formula

fdt

(vol(Ft(Nm-1)))

It=o = (div(W) - 2 - W(ln IIWII2)) - dNm-'

If, in particular, the distribution Em-' consists of the level surfaces of afunction f : Mm -' IR, and we choose the gradient of this function as thevector field, W = grad(f), then div(W) = 0(f) is the Laplacian of f.

Corollary 4. The volume change of a compact level surface N` of thefunction f : Mm R under the flow of the gradient vector field is given bythe formula

d (Vol(4 (Nm-1))) 1,=0 = JN^ 1

(°(f) - -grad(f) In ilgrad(f)I12) -dNm-'.

Remark. In all these formulas the Laplacian, the divergence, and the gra-dient are taken with respect to the manifold.

Example 7. Consider a function f : Mm - R and assume that there isanother function µ : Mm -+ IR such that

d(Ilgrad(f)112) = 2p . dfBy Theorems 4 and 5, the flow of the gradient vector field grad(f) mapslevel surfaces of f to level surfaces, and the volume change is described bythe formula

Wt(vol (41(Nm-1))) It=o = Jvm

(0(f) - µ)dNm-1

For example, the spheres S'-'(R) C R" are the level surfaces of the func-tion f (x) = IIxII2, and we obtain Ilgrad(f )II2 = 411x112 as well as 0(f) = 2m.Hence 0(f) - µ =_ 2(m - 1) is constant, and for the flow 4 (x) = e2t . xwe obtain the following differential equation describing the evolution of thevolume:

wt-

The second application of Frobenius' theorem will play an important role inthe chapter devoted to surface theory.

Theorem 6. Let 0 = (wig) be a (k x k) matrix of 1-forms defined on aneighborhood of 0 E Rm, and let AO be an invertible (k x k) matrix. In aconnected neighborhood 0 E V there exists a (k x k) matrix A = (fib) offunctions satisfying

Il = dA - A-' and A(O) = Ao


if and only ifd12 = 12 n Q.

In this case, the function matrix A is uniquely determined. If, in addition,11 is an anti-symmetric matrix (12+12' = 0), and AO is an orthogonal matrix(Ao Aa = E), then the solution A(x) is also orthogonal at each point of theset V.

Proof. First, the condition d12 = 1Z A 12 is necessary for the solvability ofthe equation Q = dA A-1. In fact, dA = SZ A implies

0 = ddA = d(f?-A) = (dIl)-A-I1AdA =The matrix A is invertible, and hence d12 = fZA11. Uniqueness of the solutionis also easy to see. For two solutions A(x) and B(x) we have, e. g.,

d(B-1) = -B-1 (dB) - B-',

and hence the differential d(B-1 . A) vanishes:d(B-' - A) = d(B-1) - A+ B-1dA = -B-1 (dB) B-1 A+ B-1 - 12 A

_ 0.

Thus B-1 A is constant, and at the point x = 0 it is equal to the unitmatrix. This implies A(x) = B(x) for all points x E V. Now we provethe existence of a solution under the condition d11 = 12 A Q. To this end,consider the following (k x k) matrix of 1-forms on the space Rm x IItk2 withcoordinates (x1, ... , x"', z') ):

k-

A =r=1

From d12 =12 A f? we obtain

dA = ddZ-d1ZAZ+flAdZ = -12A12AZ+SlAdZ= -1lAQAZ+1lA(A+12AZ) = fIAA.

kThe system of forms dz'j - E Wjrzr3 is linearly independent in IR x Rk2,

r=1and by Frobenius' theorem, there exists an m-dimensional integral manifoldMm c IRt x Ht k2 through the point (0, A0) E Ht'" x Rk2. The tangent spaceT(o,Ao)Mm to this integral manifold has only the null vector in common withRk2

TiO,AaiMm nittk2

= {0} .

This follows directly from the shape of the form A, since the tangent space toMm is determined by the equation A = 0. Then the integral manifold Mm


is the graph of a map A : W -+ Rk2 defined on an open set 0 E W C R'"satisfying the initial condition A(O) = Ao. From

A'(A) =we see that the (k x k) matrix A is the solution of the differential equationwe looked for. The remaining statements of the theorem follow from theformula

d(At A) = (dA)t A + At dA = At 52t A + At f2 - A .

In fact, if f2 is an anti-symmetric matrix, and dA = f2 A is a solution of thedifferential equation, then we immediately conclude that d(At - A) = 0. O

Exercises

1. Consider on R3 the 1-form

w= Idx+Idy+1dz.yz xz xy

Prove that the distribution defined by w = 0 is integrable. Find, moreover,an integrating factor for w, i.e. , a function f such that d(f w) = 0.

2. Let P and Q be functions in the variables x, y. u, v. Consider on RR the1-forms

wi = w-2 =

a) What are the conditions for the distribution wi = w2 = 0 to be inte-grable?

b) Let the function f : C2 --+ C be defined in the complex variables zx + iy, w := u + iv by f (z, w) = P + i Q. Prove that the integrabilityconditions for the system wi = w2 = 0 are, in particular, satisfied whenf is a holomorphic function.

3. Prove that in R3, a distribution defined by the form

w =

P Q Rdet N U = 0[PQRJ

Exercises 127

vanishes. This formula is to be interpreted in the following way: Write thedeterminant out and apply each of the resulting differential operators to therespective function.

4. Solve the initial value problem i = ex cost, x(O) = xo

a) by separation of variables;b) by finding an integrating factor.

5. The method of solving a differential equation by finding an integratingfactor is of interest, in particular, if separation of variables is impossible.As an example, solve the differential equation i(x3 - tx) = 1, x(0) = xo.How could this equation, nevertheless, be solved by a suitable separation ofvariables?

6. Solve the differential equations

a)

b) sin t + ez + cost i = 0 ;c) 4t3+6tx3 + (3t3x2 +3)i = 0 .

7. Let f = (w23) be a (k x k) matrix of 1-forms defined on an open neighbor-hood of 0 E R'. Prove that locally there exists a (k x k) matrix A consistingof functions satisfying

dA =if and only if there exists a 1-form a such that the matrix E := SZ - a Esolves the equation dE = E A E .

Chapter 5

Curves and Surfaces inEuclidean 3-Space

5.1. Curves in Euclidean 3-Space

The notion of a curve in 3-space R3 is not as simple as it may seem at firstglance. For a long time in the 19-th century, the common understandingwas that a curve is a subset A C 1[13 described in a continuous way by a realparameter. In this sense, a curve was the surjective image of a continuousmap f : [0, 1] - A C R3. To require the injectivity of f is too strong, inso far as it excluded curves with self-intersections. Apart from the fact thatthe assumption of continuity is too weak to render such a "curve" accessibleto differential calculus. a much more essential reason shattered this notionof a curve.

Theorem 1 (Peano 1890). There exists a continuous. surjective map fromthe interval onto the square.

Proof. We construct the map f : [0, 1] - 10, 1] x [0, 1] as the limit of auniformly converging sequence of continuous maps f,. In the first step, wedivide the interval [0, 1] into nine subintervals of equal length and decomposethe square accordingly. The map f, is then to map each of these subintervals,[(i - 1)/9, i/9], to the diagonal of the i-th square continuously and bijectively(see the figure). The map f2 arises in the same way when we further divideeach subinterval [(i - 1)/9,i/9] as well as each subsquare. This results in81 intervals and squares, and f2 maps the small intervals continuously andbijectively onto the diagonals of the small squares

129

130 5. Curves and Surfaces in Euclidean 3-Space

0 1/9 9/9

7 8 9

5 4

1 2 3

so that f2([(i - 1)/9, i/9]) lies in the i-th square of the first step of thedecomposition. By construction, f1(i/9) = f2(i/9). This way we obtain asequence of continuous maps satisfying

(1) II f.+1(t)II < V 2-13n for all t E [0, 11;

(2) f,,(i/3n+1) = fit+1(i/3n+1) = fn+2(Z/3n+1) for all n and i < 3n+1.

61 6 63 64 65 78 80 81

60 /5 58 6 68 s 78 7 76

55 57 7 71 73 7 75

54 52 3 38 3 36 35 34

49 51

<(

41 4 31 3 33

48 4 46 4 44 4 30 29 28

7 8 9 11 25 26 27

6 5 4 14 24 23 22

1 2 3 17 19 2 21

The estimate in (1) implies the Cauchy condition,

11f (t) - f (till <1

n r - V/2- .3n-1'

for all t E [0, 1] and in > n. Thus, the sequence fn converges uniformly toa continuous map f : [0, 1] [0, 1] x 10, 1]. By construction, all the points

(s , 3^') belong to the image of the map f. The interval [0, 1] is compact,and hence the image set f([0,1]) is again a compact subset of the square.But the subset of the image of f mentioned before is dense in the square

5.1. Curves in Euclidean 3-Space 131

[0, 1] x [0,1], and hence f ([0, 1]) = [0,1] x [0. 1], i.e. , f is continuous andsurjective.

This example had an essential impact on general topology. In fact. it showsthat the "dimension" of a topological space can increase under a continuousmap, and that this notion has to be made more precise (topological dimen-sion theory). We will not deal with this problem here, and instead confineourselves to the case of smooth curves in R3. We could try to consider theseas smooth, one-dimensional submanifolds of 1183, but that would exclude self-intersections of curves again. For this reason, we formulate the notion ofcurve slightly more generally.

Definition 1. A (parametrized) curve is a differentiable map y' : [a, b] 1<83

from an interval to 1183 whose derivative vanishes nowhere.

It is easy to see that sufficiently small image sets of a parametrized curveare submanifolds. Since dy(t)/dt 0 0, for a fixed initial point, the lengthfunction

L(t) := fat II do (u) II dp.

is monotonously increasing. A point on the curve is uniquely determinedby the parameter describing the length of the curve segment from the ini-tial point to the point considered. In other words, inverting the lengthfunction, we obtain a parametrization -y o L-1 : [0, L(y)] 1R3 of thecurve y, and the tangent vector with respect to the length parameters haslength one, IIdy/dsIj = 1. This kind of parametrization is called the nat-ural parametrization of the curve. The preceding observation shows thatthe natural parametrization can be constructed starting from an arbitraryone. We agree on the following: If the curve is given in any parametrizationy(t), then we denote by y(t), etc., the derivative with respect to the param-eter t; if the curve is given in its natural parametrization, then -y(s) is thederivative with respect to the length parameter s of the curve.

Example 1. Consider the helix in the parametrization

-y(t) = (a cos t, a sin t. b t) .

Since jjy(t)Ij = a' + 2 , the length function is L(t) = a + b2 t. Fromthis, we obtain the natural parametrization by passing from the parametert to the parameter s = L(t):

s s(s) _a2+b2 a2 +61

To every curve, we shall assign three linearly independent vectors at eachpoint of general type. The first of these vectors is the normalized tangentvector.

132 S. Curves and Surfaces in Euclidean 3-Space

Definition 2. If y(s) is a curve in its natural parametrization. the vectort(s) := y'(s) is called the unit tangent vector to the curve.

The curvature of a curve at a point measures the angle variation of thetangent vector per length unit. More precisely:

Definition 3. Let C C R3 be a curve, and let p E C be a point on the curve.The curvature K(p) of the curve C at the point p is the limit

tc(p) = lim 4q-p pq

where o is the angle between the tangent vectors at the points p and q. andpq denotes the length of the curve segment between these points.

Theorem 2. In the natural parametrization of the curve, :(s) = IIy"(s)II.

Proof. Setting p = y(s) and q = y(s + h), we obtain Tiq = h in the naturalparametrization of the curve, and the angled is computed using the formulafor the angle in an isosceles triangle,

sin(0/2) = IIt(s+h) - t(s)II/2.

Thus

lim0 = lim

0/2 2 sin(¢/2) = lim Il t(s + h) - t(s)II= I Iti (s)IIq--p pq h-.0 sin(0/2) h-0 h h-0 h

In an arbitrary parametrization of the curve, a straightforward parametertransformation yields the following formula (see Exercise 1):

K(t) = II''(t) x y(t)IIIly(t)II3

Definition 4. The principal normal vector h to a curve at a point of non-vanishing curvature is the normalized derivative of the unit tangent vector,

h(s) := 1 t'(s) = y(S)n(s) 11-y" (s)II

The vector productb(s) := t(s) x h(s)

is called the binormal vector to the curve.

Thus, at each point of the curve with non-vanishing curvature, there existthree mutually orthogonal vectors F(s), hh(s), and bb(s) all of length one, theso-called Frenet frame of the curve. Finally, we introduce a last geometriccharacteristic of a space curve, the torsion.


Definition 5. The torsion of a curve C c R3 of class C3 at a point ofnon-vanishing curvature is the scalar product

d6(s)r(s) _ -( ds

The formula for the torsion in an arbitrary parametrization is the subjectof Exercise 1.

The structural equations of a curve in euclidean space are the Prenet formu-las expressing the derivatives of the P enet frame through this same frame.

Theorem 3 (Fundamental Theorem of Curve Theory). Consider a curve ofclass C3 with nowhere vanishing curvature in the natural parametrization.Then, in the above notation,

d t(s) 0K(s)

0 t'(s)

d h(s)=

-ac(s) 0 r(s) h(s)

b(s) 0 -r(s) 0 b(s)

Let y, [0, L] R3 be two curves with coinciding curvature and torsionin their natural parametrizations. Then there exists a euclidean motion A :1R3 - R3 such that y`(s) = A o y(s). Let ac(s) be any positive function, andlet r(s) be an arbitrary function, both defined on the interval [0, L]. Thenthere exists a curve whose curvature and torsion are the given functions.

Proof. By the definition of the principal normal vector, we have t'(s) =K(s) K(s). Since 11h(s)II __ 1, the derivative k(s) is orthogonal to K(s).Hence it can be represented as a linear combination of t(s) and 9(s). Thecomputation

(dh(s), t(s)) _ (h(s), t(s)> - (h(s), j.f1> = -K(s)s

and the definition of torsion imply the second of the Frenet formulas. Butthen

d9(s) _ O(s) x h(s) + t xdh(s) = _r(s) h(s).

ds ds dsViewing the vectors of the Frenet frame as row vectors,

t(s)A := h(s)

b(s)

defines a (3 x 3) matrix A of functions on [0, L] satisfying St, wherethe entries of the skew-symmetric matrix Q of 1-forms are the curvature and


the torsion of the curve:

0 a(s) 0

Q = -K(S) 0 r(s) ds.0 -r(s) 0

Existence and uniqueness of a curve forgiven curvature and torsion functionsthus follow immediately from Chapter 4, Theorem 6, since the integrabilitycondition. dil - Q A Q = 0, is satisfied for trivial reasons. Note that thesolution of the system of linear differential equations, dA = Q A. is definedon the whole interval on which the curvature ac(s) and the torsion r(s) aregiven. Having first determined the matrix A(s) for the prescribed functionsK(s). r(s) from the equation dA = Q A, we obtain the curve -y(s) by onemore integration.

7(s) = Jt(s).ds.

Remark. The plane spanned by the tangent vector f and the principalnormal vector h at a given point is called the osculating plane of the curveat that point. The binormal vector 6 is perpendicular to this plane. Thethird Frenet formula implies Ir(s)I = 110(s)II, and hence the absolute valueIr(p)l of the torsion of a curve at a point p can be described as the limit

IrQ)I = v m

where v denotes the angle between the osculating planes to the curve at thepoints p and q (compare the proof of Theorem 2). Thus the absolute valueof the torsion measures how much the curve "winds out" of the osculatingplane.

Next we discuss some curves with special properties, and explain how theFrenet formulas can be applied to study them.

Definition 6. A curve whose tangents form a constant angle with a fixeddirection in R3 is called a slope line'.

Straight lines and helices are slope lines. These can be characterized by thefact that the quotient r(s)/tc(s) is constant.

Theorem 4 (Lancret, 1802). A curve of class C3 with nowhere vanishingcurvature is a slope line if and only if r(s)ln(s) is constant.

Proof. If there exists a vector a E R3 such that (t(s), a) is constant, then,differentiating this equation, we obtain that the scalar product (h(s). a)

1 In German. such a curve is called Boschungslinie.


vanishes, since K(s) # 0. Differentiating once again leads, using the Frenetformulas, to

T(s) (6(s), a = c(s) .

The vector d lies in the if, 9}-plane, and (t, a) is constant. Hence (9, d) isalso constant, and thus the quotient T/rc is constant. Conversely, if r/K isconstant, then consider the vector

T t(s) + 6(s).K

The Frenet formulas imply

ds I£ ds + ds =T h - T h = 0,

and hence a is a constant vector. The scalar product (t(s), d) = T/rc is alsoconstant, and the curve is thus a slope line. O

Curves lying on a sphere in 3-space can be described by a similar relationbetween curvature and torsion.

Theorem 5. A curve of class C4 with nowhere vanishing curvature andtorsion, K, T 96 0, lies on a sphere of radius R > 0 if and only if it satisfies

I+

,i 2_ 2= RK2 rc2T

in the natural parametrization.

Proof. Differentiating the equation II-'(S)II2 ° R2, we obtain (t(s), -Y(s)) _0, and hence y(s) is a linear combination of the vectors h(s) and 6(s),

y(s) = a(s) h(s) +,13(s) b(s) .

Furthermore, IIy(S)112 - R2 implies a2(s) + f32(s) R2. Differentiating(t(s), y(s)) - 0 again leads to K(s) (h(s), y(s)) + 1 0, and thus

a(s) =

We differentiate the equation c(s) (h(s), y(s)) + 1 - 0 and obtain thefollowing relation by a simple transformation:

ic'(a(3)

2s)I£ (S)T(S)

The asserted necessary condition for a spherical curve then immediatelyfollows:

1 (r 2

R2 = a2(s) + $2(s) =a2(s) +

K2(3)T(S)/


If, conversely, this relation between curvature and torsion holds for a C4-curve, then we first differentiate it and obtain the equation

r(s) d KG(s) _/G(S) ds !GZ(s)r(s)

0.

Now consider the vector

a(s) := 7(s) +;R-S)

h(s),z() (s)

.6(s).

Using the Frenet formulas and the preceding relation between curvature andtorsion, we compute the derivative of the latter, and find that

id(s) = 0.Hence d:= a(s) is constant, and

11-y(S) - aI1z = az(s) + \az(s)(s)/z = Rz'i. e., the curve -y(s) lies on the sphere of radius R with center d. 0

Next we turn to plane curves. Note first that these can be described as thecurves with vanishing torsion.

Theorem 6. A curve of class C3 with nowhere vanishing curvature, k(s)0, lies in a plane in R3 if and only if its torsion r(s) - 0 vanishes identically.

Proof. Let a' be a vector perpendicular to the plane in iR3 containing thecurve. All the tangent vectors t(s) lie in this plane; hence (t'(s), a) - 0.Since ac(s) # 0, we immediately obtain (h'(s), a") - 0 by differentiating thisequation. Thus d coincides with the binormal vector 6(s). In other words,the binormal vector 6(s) is constant. Then 0 = 6'(s) = -r(s) h(s) impliesthat the torsion vanishes, r(s) - 0. The converse is proved analogously. 0

The curvature of a plane curve can be ascribed a sign. In fact, the principalnormal vector is proportional to the vector obtained by rotating the tangentvector through the angle 7r/2 in the positive sense. The curvature of a planecurve is ascribed the positive sign if the corresponding factor is positive.Identifying R2 with the complex numbers, the rotation through 7r/2 in thepositive sense corresponds to multiplication by the number i E C. Using themultiplication of complex numbers, this leads to

Definition 7. Let -y : [0, L] - C = R2 be a plane curve in its naturalparametrization. The plane curvature k(s) is the function k : [0, L) -' Rdefined by the equation

ast(s) = k(s) i t(s) .


The absolute value jk(s)j of the plane curvature coincides with the curvatureK(s) of the curve viewed as a space curve.

Example 2. -y(t) = (t, ±t2) .

A closed curve i' : [0, L] - C = R2 is one that starts and ends at the samepoint and whose tangent vectors at this point coincide as well, i'(0) = -Y(L)and t(0) = F(L).

Theorem 7. Let [0, L] C be a closed curve. Then the integralr L

2 Jk(s)ds = 2

IL7

is an integer, called the winding number of the closed curve.

Proof. Consider the map t' : [0, L] C defined by

t* (s) = exp [k(u)du]

Then dt'(s)/ds = i k(s) - t*(s). and hence (t(s)/t*(s))' = 0. The tangentvectors t(s) are thus described by the formula

F(s) = C exp Lk(u)duj

fLfor a certain constant C. Since t(0) = t(L), the number J k(s)ds is anintegral multiple of 27r.

o 0We conclude the section on curves with the discussion of the Fenchel in-equality, which claims that the total curvature of a closed space curve is


bounded from below by 2;r. We start by considering plane curves, and thengeneralize the result to space curves. First we need an auxiliary observation.

Lemma 1. Let cp : [a, b] lR be a real function of class C', and suppose thatthe function f (t) := ew<<l" joins the numbers ±1, i. e., f (a) = 1, f (b) = -1.Then

(1) f If(u)Idµ 2: ir;ba

(2) in this estimate, equality holds if and only if the derivative cp doesnot change sign, and, moreover, IV(b) - p(a)I = r.

Proof. From j (p) = i we obtain

fb

If (j)I dµ ? I fb

(p)dui = Iw(b) - w(a)I.

Since f (a) = 1 and f (b) = -1, there exist integers k, I such that W(a) = 2kirand yp(b) = 21;r + tr. This implies

af bIf(i)Idy > irI1+2(k-l)I > a.

The case of equality follows immediately. 0Theorem 8. Let -y : [0, L] - R2 = C be a closed plane curve. The totalcurvature of the curve is at least 27r:

jLjIk(s)Ids = > 2ir.

Equality in this estimate occurs if and only if the closed curve -y bounds aconvex region.

Proof. To every point of the curve we assign the tangent vector to the curve.This determines a map from the interval to the unit circle S' satisfyingF(0) = F(L). The image set F([0, L]) C S' is a compact and connected subsetof S', hence an arc. This arc contains at least one pair ±zo of oppositepoints. Otherwise, there would exist a number zo E S' such that neither+za nor -zo would belong to the arc. But then o F(s) would have animaginary part which would have constant sign, and from

F(s)1 ds = Im\ a

f L F(s)ds) = Im \ (y(L) - y(0))) = 0J:Im aG - Z Z;

we would obtain a contradiction. Without loss of generality, we may assumezo = I and r(O) = F(L) = 1. Moreover, there exists a parameter value


Si E [0, L] such that r(s1) _ -1. Applying the lemma above, we obtain0sl 81 L Lf ik(s)Ids = f IZ'(µ)Idi >- 7r, f Ik(s)Ids = f 71,

0 81 81

L

and, altogether, JIk(s)Ids >_ 2a. The inequality for plane curves is thus

proved. In the case of equality, the lemma implies that the plane curvaturek(s) of the curve does not change sign on the interval [0, s1], and

/081 Lk(s)dsI = it = 14L k(s)dsl .

Now we exclude the case that k(s) has different signs on the intervals [0, s1]and [s1, L]. If this were the case, then

k(0) = k(si) = k(L) = 0,

and k(s) would be positive on the interval [0, s1] and negative on [s1, L].But then the inequality

k(µ)dp < 7r0 < fwould hold for all parameters s E [0, L], and the tangent vector t(s) _

expy

k(p)dµ] would stay in the upper half-plane (z E C : Im

hence we derive a contradiction toL

t(s)ds = -t(L) - ^t(O) = 0.

The plane curvature k(s) of the curve thus has constant sign, k(s) > 0. Inthis case, the whole curve completely lies on one side of the tangent to thecurve at an arbitrary point. In fact, without loss of generality, we can choosethe point as that corresponding to the parameter value s = 0. Consider thefunction /3(s) measuring the height of the curve point -y(s) over the tangentthrough -t(0),

Q(s) := Re (('Y(s) - Y(0)) (i t(0))

The derivative of Q(s) is easily computed:

13'(s) = Re 00) exp I i f e k(u)du] Ti t(0))) = sin(j8

k(u)du)L o

The curvature is non-negative, and thus /3(s) is negative for parameter valuesclose to s = 0. If /3(s) had any negative value in the interval [0, L], thenthere would exist a value snin such that

/3(smin) < 0 and Q'(smin) = 0.


/LSince J k(u) du = 21r, this value s is determined by the equation

05min

k(u) du = 7r,

and, for s E [0, smin], the derivative of the function 0(s) would be negative.Therefore, $(s) would not be decreasing in [0, smi,,], contradicting /3(smio) <0. Summarizing, we obtain that the height function is non-negative, andthus the curve lies completely on one side of each of its tangents. Considernow the half-planes determined by all the tangents. Their intersection is aconvex domain whose boundary coincides with the curve ry(s). 0

We generalize the inequality of the preceding theorem to space curves. Todo so, we need a preparation.

Lemma 2. If D C S2 is a closed curve in the sphere of radius one whoselength does not exceed 2a, then V is contained in a hemisphere.

Proof. Choose two points P and Q on D which divide the curve into twosegments of equal length, V = D1 UD2. Let the north pole N = (0, 0, 1) lieon the shorter segment of the great circle through P and Q on the sphere.Denote by Di the curve segment obtained from Dl by rotation around thez-axis through the angle a. The union Dl U Di is a closed curve on thesphere whose length coincides with that of the original curve D. If the curvesegment Dl intersects the equator S' = {(x, y, 0) : x2+y2 = 1} of S2, thenthe curve Dl U DI contains opposite points on the sphere. Hence its lengthhas to be at least 27r. By assumption this length cannot exceed 27r. Thus 7)idoes not intersect the equator S' at all, if the length of the curve D is smallerthan 27r. Otherwise, D, or V2 is a semi-circle on the equator, respectively.In both cases, the curve D is contained in the upper hemisphere. 0

Q

5.2. The Structural Equations of a Surface 141

Theorem 9 (Fenchel Inequality). The total curvature of a closed curveC C R3 is at least 27r,

JcK(s) ds > 27r.

Equality holds if and only if C is a plane curve bounding a convex domain.

Proof. Let a non-plane curve be given. Consider its tangent map t :[0, L] -+ S2 as a "curve" in the sphere S2. The length of the latter co-incides with the total curvature of the original curve,

fL rLL(t) = I

J0 o

Note first that the image of t cannot be contained completely in a hemisphereof S2. If this were the case, then there would be a vector a' E S2 such that(a, F(s)) _> 0 for all values of the parameter. But the integral of t(s)vanishes, since we started from a closed curve. Hence (d, F(s)) - 0, andthe curve C would be a plane curve. The assertions of the theorem nowfollow from the previous lemma together with Theorem 8.

5.2. The Structural Equations of a Surface

A surface is a two-dimensional submanifold of 1R3. As in the case of curves,we want to allow for self-intersections and will also consider parametrizedsurfaces. This is meant to denote a map F : U --+ R3 defined on an opensubset U of R2 (or, more generally, on a two-dimensional manifold U) withthe property that the differential D(F) has rank two at each point. As inthe case of a two-dimensional submanifold A12 C l3, we denote by M :M2 - R3 the inclusion into 3-space and view this map as a vector-valuedfunction. Locally, we choose an orthonormal tangent frame el, e2 to thesurface and denote by e3 := el x e2 the normal vector to the surface. Wesometimes consider-without introducing a new notation-el, e2 and e3 alsoas vector-valued functions on the manifold 1112. Denote by al and o2 theframe of 1-forms dual to {el, e2}. The differential dM of the identity map onthe surface is the identity map on each tangent space to the surface. Hencethe following equation holds:

dM =01 - el +

expressing nothing but the decomposition of a tangent vector with respectto the basis e1, e2. The exterior product of A o2 is a 2-form on the surfacewhich is independent of the choice of the orthonormal frame el, e2 for a fixedorientation and coincides with the volume form,

012 = of A a2.


The scalar products (ei, ej) of the vector-valued functions ei are constantfunctions on M2. By differentiation, we obtain

(dei, ej) + (ei, dej) = 0

and denote the 1-forms by wij := (dei, ej). Now combine the differentialforms into an antisymmetric (3 x 3) matrix Cl:

0 w12

= w21 0W31 W32

W13

W23

0

Using the forms just introduced, the differentials of the vector-valued func-tions ei can be represented as

3

de. = E wij ejj=1

Altogether, we constructed five differential forms of degree one, the formsof and wij. These are, however, not independent; the so-called structuralequations of a surface express the relations among them. The fact that theform w12 is completely determined by 01 and 02 will turn out to be of specialimportance. Later on, this will imply the fact that the Gaussian curvaturebelongs to the inner geometry of a surface (Theorema Egregium, Gauss,1827).

Theorem 10 (Structural Equations of a Surface).

(1) d01 = W12 A C2, do2 = w21 A o1 .The form W12 is completely determined by the forms of and o2 .

(2) a1 A w13 + 02 A w23 = 0 .

(3) dC = n A Cl .

Proof. Differentiate the function M twice,

0 = ddM = d(o 1 - el + 02 e2) = (dal - w12 A 02) el

+ (do2 - w21 A 01) e2 - (a A w13 + 02 A w23) e3 .

This yields the equations stated in (1) and (2). The identity for Cl followsusing 0 = ddei = Ek (dwik - Ej wij A wjk) ek. The form w12 can becomputed from al and 02 as follows. Represent the differentials

d01 = A 01 A o2i d02 = B 01 A o2

as multiples of the 2-form of A 02. The first structural equation implies

w12 = Aa1+Bo2.


Let a parametrization F : U - M2 C R3 of a surface by a a domainU C R2 be given; then the 1-forms ai and wij can be pulled back to U. Theresulting forms will be denoted by a, := F'(ai) and wij := F'(wij). Thenthe structural equations of the surface hold as well:

2

do,; _ wij A aj*, al A W13 + a2 n w23 = 0, dtl' = St' n it .

j=1

Now we prove a first formulation of the fundamental theorem of surfacetheory. This allows to determine a surface locally in 3-space from the systemof associated 1-forms satisfying the structural equations relating them.

Theorem 11 (Fundamental Theorem of Surface Theory-First Formula-tion). On a simply connected, open subset U of R2 let four differential forms01, a2, 013, 023 of degree one be given. Suppose that the forms 01 and 02 arelinearly independent at each point. Define the 1 -form 012 by the equations

d'1 = 012/02, d02 = -012A01.Extending them antisymmetrically, wji := -wij, assume that the given sys-tem satisfies the structural equations

U1 A 01$ + a2 A 023 = 0, dfl = NA N.Then there exist a parametrized surface F : U -' R3 and an orthonormalframe of tangent vector fields such that the induced forms a, and wij coincidewith the original forms aFi and wij, respectively. The surface together withits orthonormal frame is uniquely determined up to a euclidean motion ofR3.

Proof. The (3 x 3) matrix S2 is skew-symmetric and satisfies the integrabilitycondition dSl = SE A li. For a given initial condition Ap E SO(3, R), thereexists a matrix q of functions such that St A = dA by §4.3, Theorem 6. Therows of this matrix are orthogonal and have length one. Thus they definethree vector-valued functions e1, e2, e3 : U - R3, and

3

dei =j=1

Consider the 1-form0 := 01.91 + 02.92

Computing the differential of 0 yields

dO = 0.

By Poincares' lemma (§3.6, Theorem 22) there exists a map F : U - R3 suchthat dF = ¢, = Q1 e1 +02 e2i and hence we obtain the parametrized surfacetogether with the orthonormal frame. Twice in this proof we integrated a


differential equation, and thus there remains a certain degree of freedomwith respect to the initial condition. First, the solution q of the differentialequation Il A = dA depends on the prescribed orthogonal matrix Ao ESO(3, R). In the second step, the function F : U -' R3 is determined bythe differential equation dF = ¢ only up to a constant vector a E 1R3.Summarizing, we see that the surface is, for prescribed forms of and wtj,determined up to a euclidean motion.

Example 3. We discuss the case of a surface of revolution in R3. Considera plane curve in the (x, z)-plane with component functions r(s) > 0 andz(s). The surface F arises by letting this generating curve revolve aroundthe z-axis, and thus it has the following parametrization defined on U =1[2 x (0, 27r):

F(s,cp) = (r(s)sincp, r(s)cosco, z(s)).

For simplicity, we suppose that the generating curve of the surface F isparametrized by arc-length, i.e., at each point the following condition issatisfied:

(*) (r')2 + (Z,)2 = 1.(Compare Exercise 15 for the general case.) First we compute two tangentvectors to the surface,

aas = (r sin co, r cos co, z'), a = (r cos V, -r sin tp, 0),

as well as their scalar products,

a'

s=

(r,)2+ (z,)2 = 1,

a,

s0,

a s-r2as as as 5 = app asp -

Normalizing the second tangent vector correspondingly, we obtain the or-thonormal frame

_ a _ 1 ael as, e2 r Op

and the basis dual to it,

01 = ds, 02 = r dip.

The unit normal vector to the surface F is the vector product of el and e2,

z'sin,pe3 := el x e2 = z'cos

-r'We compute the differential of el = (r'sin gyp, r' cos cp, z') componentwise,

del = (r" sin cp ds + r' cos cp dcp, r" cos pds - r'sin pdcp, z"ds),


and check that

del = r' dcp e2+r

I ds e3 .z

In fact, inserting e2 and e3 into this linear combination, we obtain

r' dcp e2 + ; ds e3z

= (r' cos cp dcp, -r'sin cp dcp, 0) + z, (z'sin cp dcp, z' cos cp ds, -r' ds)

= (r" sin cpds + r' cos cp dcp, r" cos cp ds - r'sin cp dcp, -r" r'/z' ds) .

But, since differenting the relation (*) implies rr' + z'z" = 0, this is equalto del. Similarly, one proves that

de2 = -r'4 el - z' dcp e3, de3 =-r

ds el + z' dcp e2,z

and, therefore, we computed the matrix f of 1-forms w, completely:

0 W12 w13 0 r dcp Z' ds-w12 0 w23 = -r dcp 0 -z'dp-w13 -w23 0 : ds z' dcp 0

In addition, we show in this example how to compute the form w12 directlyfrom al and a2. Starting from dal = 0 and dal = r'-dsAdcp = r/r-al Aa2,we obtain the formula

W12 = r -a2 = r - dtp

using the method described in the proof of Theorem 10. Computing w12directly from the definition turns out to be even easier:

r'w12 = (del, e2) = (r'dcp e2 + z'

ds e3, e2) r' dcp -

Example 4. Let h : U R be a real function in two variables on the openset U C R2, and denote by F its graph,

F(x,y) = (x, y, h(x,y)).

We indicate partial derivatives by lower indices. Starting from the twotangent vectors

ax = (1, 0, h=) anda

= (0, 1, hy),ay

we determine an orthonormal frame with the same orientation,

1 a hZhy y 1+ bye, e2 _ a- a= l lJ-+h2 x+ h2 1 + h2 + h2 h,hy ay ax


as well as the 1-forms dual to these vectors,

hxhy VT T h2 + h2of = 1+h2dx+

1+h2 dy' Q2 = l+h dy.2

The corresponding volume element is

of Aa2 = 1+hz+hndxndy.

The computation of the whole matrix fl is more extensive. Because of itsgeometric relevance, we give the form w12 for later use:

W12 = (del, e2) =hu dl

(1+h2 1+h2+h2

In the discussion of curvature properties of surfaces, we will see that partic-ularly interesting graphs arise from holomorphic functions. Therefore, theseare discussed next.

Example 5. Let f : U C be a holomorphic function. The modularsurface of f is the graph of the modulus function h = if I = (fl) 112 of f.The relations between the derivatives with respect to the complex parameterz and the (real) partial derivatives are expressed in the following formulas:

8 1 a 2.) , 8-

1 8 88z - 8x - z by 8z 2 ex + i FY)

Hence it is possible to express hx and by by the z-derivatives of f and f.The latter are denoted by '. It is easy to see that

8h f'f 8h ff'8z

_]F h-' 8z

_2h

Thus the derivatives we wanted to compute are

hz = 2h(f'f +ff) and by =Rewriting this yields, e. g. ,

1+h2+h2 = 1+1992,and inserting these expressions into the formula of the previous example weimmediately obtain an explicit expression for the corresponding form w12depending on the complex derivatives of the function f. Indicating the linesof constant modulus as well as those of constant argument on its modularsurface leads to a clear picture of the behavior of the function in the complexdomain. As an example, we consider the function f (z) = 1/ sin(z) outsideits poles, shaded according to the argument and showing the level lines ofthe modulus:

5.3. The First and Second Fundamental Forms of a Surface 147

Example 6. Decompose a holomorphic function f : U - C into its realand imaginary parts, f = u + iv. Taking in Example 4 for h the functionsh = u and h = v, respectively, we obtain the graphs of the real and theimaginary part of f. Starting from the Cauchy-Riemann equations,

ux = Vy, uy = -vx ,and first using the formula for 8182 derived in the preceding example, f _ux - iuy = vy + ivx, we obtain

1+uz+uy = 1+vv+v22 = l+If'I2.At this point we omit the explicit formula for w12.

5.3. The First and Second Fundamental Forms of a Surface

In classical differential geometry, it is common to describe a surface by twosquare matrices, the so-called first and second fundamental forms. Thefirst of these is already familiar from Chapter 3, since the first fundamentalform is just the Riemannian metric. The second fundamental form dependson the particular realization of the surface in euclidean space. To be ableto define the first fundamental form in the language of differential forms,


we introduce the symmetric product of 1-forms, in addition to the exterorproduct we already know.

Definition 8. Let M' be a manifold, and let a, µ be two 1-forms on Mn.Define a symmetric bilinear form a 0 p : TM" x TM" IR by

a 0 µ(V, W) := 2 (a(V)µ(W) + a(W)µ(V)) .

In the literature, it is common to omit the symbol 0. The symmetric squareof a 1-form or is often denoted by a2.

The definition immediately implies that, on a surface with orthonormalframe el, e2 and corresponding dual basis of 1-forms al, a2, the followingequation holds:

a: O al(ej, ee) = ai(ei)a1(et) = bijb k

For the basis e1, e2 the bilinear form

I := al 0ol+a2Oa2is thus precisely the usual euclidean scalar product. It is called the firstfundamental form of the surface; it coincides with the Riemannian metricdefined in Chapter 3. Geometric quantities defined for the surface dependingonly on the first fundamental form are called quantities belonging to theinner geometry.

Example 7. For the surface of revolution from Example 3, we had al = dsand a2 = r dip; thus the first fundamental form is

I = ds O ds + r2dcp O dcp.

This is also written as I = ds2 + r2dc 2.

Example 8. The first fundamental form of a graph (Example 4) is

I = (1 + h2) dx2 + 2 hhxhy dx dy + (1 + h2) dy2 .

Example 9. On the upper half-plane x2 = {(x,y) E R2 : y > 0}, weconsider the 1-forms

dx dyal = -, a2 = -.y y

They determine the Riemannian metric

dx2 + dyey2

The half-plane N2 endowed with this metric is called the hyperbolic plane.The form w12 depending only on al and a2 is equal to al.


If V and W are two tangent vector fields to the surface M2, then, viewingW as a vector-valued function and differentiating it as such with respect tothe vector field V, we denote the resulting vector-valued function by V(W).In general, V(W) is not tangent to the surface. However,

Theorem 12. Let V and W be two tangent vector fields on a surface. Thenthe difference V(W) - W(V) is also tangent to the surface, and it coincideswith the commutator of the vector fields,

V(W) - W(V) = [V, W] .

Proof. In the proof we make use of the structural equations of the surface.V and W are tangent vector fields, and hence the scalar products (V, e3) _(W, e3) = 0 vanish identically. This implies

(V(W) - W(V), e3)

_ - (W, de3(V)) + (V, de3(W))

- (W, w31(V)el +w32(V)e2) + (V, w31(W)el +w32(V)e2)

al A w31 (V, W) + a2 A W32 (V, W)

Because of the structural equation a1 A w31 + a2 A w32 = 0, the differenceV(W) - W(V) is a tangent vector field on the surface. Moreover, from §3.9and the structural equation, we conclude for i = 1, 2 that

a;(V(W) - ),V(V)) = (V(W) - W(V), e;)= V((W,e1)) - (W,de1(V)) - W((V,e1)) + (V,de;(W))

2

= V((W,e;)) - W((V,e:)) +1: aj Awii(V,W)i=1

= V(a;(W)) - W(a2(V)) -dai(V,W) = a+([v,WI).Thus V(W) - W(V) coincides with the commutator [V, W]. 0We decompose the vector-valued function V(W) into its tangent and itsnormal part. The tangent part is denoted by V W, whereas the normalpart will give rise to the so-called second fundamental form II(V, W) of thesurface.

Definition 9. Let V and W be tangent vector fields on the surface M2 C R3.The tangent part of V(W),

V W := V(W) - (V(W), e3) e3 = V(W) + (W, de3(V)) e3,

is called the covariant derivative of the vector field W with respect to thevector field V. The normal part of the vector-valued function V(W) will bedenoted by II(V, W),

II(V, W) := (W, de3(V)) :


it is called the second fundamental form of the surface.

Remark. The second fundamental form of a surface depends on the ori-entation of M2. Changing it reverses the normal vector e3, and hence IIchanges sign, too.

First we collect the properties of the covariant derivative V.

Theorem 13.

(1) Vv(W1 + W2) = Vv(W1) + Vv(W2);

(2) Vv,+y,W = Vy,W+Vv,W;

(3) If f : M2 R is a smooth function, then

Vv(f W) = df (V) W + f VyW and V1.yW = f VyW;

(4) V (W1, W2) = (VVW1, W2) + (W1, VvW2);

(5) VyW - VwV = [V, W).

Proof. The additivity of VvW is a straightforward consequence of the defi-nition. The last formula follows from the previous theorem. All the remain-ing assertions are easily checked, e. g.,

Vv(f - W) = V(f W)+(f - W, VyW.

Representing the vector field W in an orthonormal frame e1, e2 on the sur-face, W = W' e1 + W2 e2, and taking into account that

VVeI = del(V) + (el,de3(V)) e3

= w12(V) . e2 + w13(V) e3 + wsi(V) e3 = w12(V) e2,

we obtain the important formula

VvW = (dW' (V) + w21(V) W2) - el + (dW2(V) + w12(V) W') e2 .

The form w12 depends only on a,, a2i and hence the covariant derivativeVvW also depends exclusively on quantities belonging to the inner geom-etry of the surface. The second fundamental form lI(V, W) is a symmetricbilinear form on the tangent bundle of the surface.

Theorem 14. Let V and W1, W2 be tangent vector fields on a surface, andlet f1, f2 be smooth functions. Then

(1) the second fundamental form is symmetric, II(V, W1) = II(Wi, V);

(2) II(V, f1 Wi + f2 W2) = f1 II(V, Wi) + f2 II(V, W2).


Proof. We prove the symmetry of the second fundamental form; the re-maining property immediately follows from the definition. With V and W,the commutator [V, W] = V(W) - W(V) is a tangent vector field as well.Thus

(V, de3(W)) (W(V), e3) = - (V(W), e3) + (V(W) - W(V), e3)

_ -(V(W), e3) = (W, de3(V)),i.e. , II(V, W) is symmetric.

Because of the structural equation de3 = W31 el + W32 e2, the secondfundamental form can be written as a symmetric product of the following1-forms:

II = W31 O Q1 + W32 O v2.

Hence, in the orthonormal frame el, e2, the second fundamental form isrepresented as the symmetric matrix

II = I II(e1, el) II(el, e2)1II(e1,e2) II(e2,e2)

w31(el) (31(e2)

w32(el) w32(e2)]

If, on the other hand, F : U - M2 C R3 is a parametrization of the surface,and we choose the normal vector

_ F X OF - 1 8Fe3 II e-Fr i

OF11

Ig-(OF

Oy1X Oy2)

57then the second fundamental form is given by the symmetric (2 x 2) matrix

b _ b11 b12

b21 b22,

with coefficients

_ 82F 1 82F OF OF, e3) -

I W2/

So for each surface M2 C R3, there exist two symmetric bilinear forms, I andII, of which the first is positive definite at each point. These two fundamentalforms are not independent; the structural equations of the surface will leadto a pair of differential equations relating them to each other. In order toformulate these equations, we introduce the curvature tensor of the surface.This is a transformation R : TM2 x TM2 x TM2 - TM2 associating withevery triple of vectors a tangent vector.

Definition 10. Let U, V, W be tangent vector fields on the surface. Thecurvature tensor R is defined by the following formula:

R(U, V)W := VuvyW - VvvuW - vu,vlW .


The following relations for the curvature tensor are a formal consequenceof the properties of the covariant derivative VvW for vector fields stated inTheorem 13. We leave their verification to the reader.

Theorem 15. Let U, V, W be tangent vector fields, and let f be a junction.Then

(1) R(U,V)(f'W) = f.R(U.V)W.(2) R(U, V)W = -R(V,U)W,

(3) I(R(U, V)W1, W2) = -I(R(U, V)W2, W1),

(4) I(R(U, V)W1, W2) = I(R(W1, W2)U, V),

(5) 1(U.V)W+7Z(V,W)U+7Z(W,U)V = 0.

Recalling that the covariant derivative VvW of two tangent vectors onlydepends on the forms o1, o2 belonging to the inner geometry, the same alsoholds for the curvature tensor. We prove a local formula for R(U, V)Wspecifying this fact.

Theorem 16. Let e1, e2 be an orthonormal frame on M2. Then

R(U,V)W = dw12(U, V) (°t(W) e2 - o2(W) e1) .

Proof. Because of the symmetry properties of the curvature tensor statedin the preceding theorem, it suffices to prove this formula for the basis fieldsU = Cl. V = e2, W = e1. In this case,

1Z(e1,e2)e1 = Vei(Ve2e1) - Ve2(Veiel) - V[eI.e2]e1V'. (w12(e2) . e2) - Ve2(u12(el) ' e2) - W12([el e2]) - e2

(el(w12(e2)) - e2(w12(el)) - W12([el,e2]))'e2

+ (wt2(e2)W21(e1) - w12(et)w21(e2))'el

dw12(el,e2)'e2,

and this last expression coincides with the right-hand side of the equationto be proved.

The first fundamental form is positive definite. Hence the symmetric secondfundamental form can be represented by a symmetric endomorphism ST A,12 - TAP defined by

II(V, W) = I(V, S(W)) ,

the so-called Weingarten map of the surface (see §5.4). The equation II(V, W)(V. de3(W)) implies that S can be rewritten as

S = de3 = W31 ' el + w32 e2 .


The covariant derivative of an arbitrary endomorphism E : TM2 - TM2of the tangent bundle is a 2-form VE : TM2 x TM2 TM2 with values inTM2. This is defined by the equation

VE(V, W) :_ Vv(E(W)) - Vw(E(V)) - E([V. W]).

Theorem 17 (Gauss and Codazzi-Mainardi Equations).

(1) VS = 0 (Codazzi-Mainardi equation);

(2) IZ(U, V)W = II(V, (Gauss equation).

Proof. Using the formula for the exterior derivative of a 1-form, the equa-tion dS = dde3 = 0 first implies

0 = V(S(W)) - W(S(V)) - S([V, W]).Looking at the tangent parts and taking into account that S([V, W]) isalready tangent, we obtain

0 = v (S(W)) - Vw(S(V)) - S([V, W]).Hence VS = 0 is proved. Recalling that the second fundamental formis II(V. W) = (V, de3(W)), we obtain S(U) = de3(U). Using now de3 =w31 el +W32 - e2 and then applying the preceding theorem proves the Gaussequation:

II(V, W) S(U) - II(U, W) S(V) = w13 A w32(U, V) (o l (W) e2 - el )

= dw12(U,V) (al (W) e2 - o2 (W) e1)= R(U, V)W. 0

Next we will prove that prescribing a positive definite first fundamental formas well as a second fundamental form uniquely determines a surface up to aeuclidean motion in R3, provided that both fundamental forms satisfy theGauss and the Codazzi-Mainardi equations.

Theorem 18 (Fundamental Theorem of Surface Theory-Second Formu-lation). Let two symmetric bilinear forms 1, II : TU x TU -+ R be given ona simply connected, open subset U of R2, and suppose that the first, 1, ispositive definite at each point. Starting from 1, the equation

21(VUV, W) = U(1(V, W)) + V(1(U, W)) - W(I(U, V)) + 1([U, V]. W)

+1(V, [W, 141) - I(U, [V, W])

defines a covariant derivative V for vector fields with respect to vector fields,and the curvature transformation is determined by

R(U,V)W := VuvvW - VvVuW - v[U,V]W.

If the symmetric endomorphism S : TU TU defined from II by II(U, V) _I(U, S(V)) satisfies


(1) V S = 0,

(2) R(U, V)W = II(V, W) -3(U) - II(U, W) S(V),

then there exists a parametrized surface F : U -+ R3 such that the inducedfirst and second fundamental forms of this surface coincide with I and II,

1 = F'(I), II = F'(II) .

This parametrized surface is uniquely determined up to a euclidean motionof 1113.

Proof. The proof is based on a straightforward application of the funda-mental theorem of surface theory in its first formulation (Theorem 11, §5.2).To apply this, we choose a frame {ei, e2} of vector fields on the open setU C JR2 which is orthonormal with respect to the bilinear form I, and de-note by 01, 02 the corresponding dual frame of 1-forms. Define additional1-forms by the equations

w12((V) := Wvei, e2),

013(V) 11(V, e1) = 1(S-(V), el),

i23(V) := R(V, e2) = 1(3(V), e2)-

This leads to a system {ai, Q2, "12, W13, X23} of 1-forms, which is extendedby requiring antisymmetry, wji := -Oil. The equations we supposed tohold imply that the integrability conditions of the fundamental theorem ofsurface theory in its first formulation are satisfied. To see this, note firstthat the equation defining the covariant derivative V immediately yields theformulas

U(I(V, W)) = I(VuV, W) +I(V, VuW), Vuv - Vvu = [u, V] .

From these, we obtain

doi(U,V) = U(o1(V)) - V(aF1(U)) -aFi([U,v1)

= U(I(V,e1)) -V((U,e1)) -I([U,V],ei)= I(VuV - VvU - (U, V), ei)+1(V, Duel) - I(U, Vve1)

= w12(U) . a2(V) - w12(V) U2(U) = 012 AQ2(U,V)

Hence dv1 = 012 A o2, and similarly one verifies that dv2 = 021 A U1. Inthe next step, we show that v1 A 013 + Q2 A 023 = 0. This follows from

01 A 013(U, V) + Q2 A O23(U, V) = II(V,U) - II(U, V) = 0,

since the second bilinear form is symmetric. For the computations to follow,note that I(ei,ei) e 1 implies I(Vu'el,'e1) = 0, and hence Duel is parallelto e2. Thus for arbitrary vector fields U, V we always have

IMue1, Ove2) = 0.

5.4. Gaussian and Mean Curvature 155

Using this equation and the Gauss equation, we compute the form dw12:

dw12lu,V) = U(i(vvel,e2)) - V((VUel,e2)) - I(V[u,v]el,e2)= I(R(U, V)el, e2) = II(V, el) II(U, E2) - II(U, E1) . II(V, e2)

= w13Aw32(U,V)

From this equation we get the structural equation to be satisfied, d012 =w13 A 032. The relation for d013 is derived in a similar way using theCodazzi-Mainardi equation:

d0134 V) = U(I(S(V),el)) -V((S(U),el)) -1(3([U,V]),El)= I(V S(u, V), el) + I(S(V), duel) - I(S(U), vvel )

= I(S(V), F2) I(e2, 7ue1) - I(S(U), e2) I(e2, vvel)

= 012A023(U,V)Thus d013 = 012 A 023, and an analogous computation yields d O23 =w21 A W13. Together, the system of forms {Q1, a2, w12, 013, w23} satisfiesthe integrability conditions of Theorem 11, §5.2, and hence the existence ofa parametrized surface F : U 1R3 such that F'(I) = I and F'(II) = IIis proved. The uniqueness statement is a consequence of the correspondinguniqueness result of that theorem. 0

5.4. Gaussian and Mean Curvature

Consider a surface M2 C R3 and its first fundamental form, the inducedRiemannian metric. Choosing a local orthonormal frame {el, e2} with dualframe al, a2, we know that the 1-form w12 = (del, e2) = (Vel, e2) is com-pletely determined by the forms al, 02. Since its exterior differential dw12 isa 2-form, there exists a function G on the surface such that

d012 =

The function G is independent of the choice of the local frame of vectorfields. Every other frame {ei, e2} can be represented as

ei = e2,

e2 = -µ sinap el +coscp e2,where µ = ±1 is constant, and V is a function on that part of M2 whereboth orthonormal frames are defined. The dual frames are thus related by

d = µ cos 92 of + sin cp o,2,

a2 = -µ sin p a2 + cos a2,

and we obtain ai A o2 = µ of A a2. On the other hand, the form w12 is equalto w12 = (dei, e2) = µ w12 + dip, leading to the formula &.012 = µ &012-This observation shows that the function G is uniquely defined.


Definition 11. The Gaussian curvature G : M2 -* R is the function definedby the equation

dw12 = - not.Remark. The Gaussian curvature depends exclusively on the first funda-mental form of the surface. If this is given, we can choose an orthonormalframe of 1-forms al, a2. From the equations

dal = A- al na2 and da2 = B - al n ag,

we determine two functions A, B and then introduce the 1-form

wit =Finally, the defining equation for the Gaussian curvature is

dw12 =

This way, the Gaussian curvature can be computed in practice.

The formula for the curvature tensor of the surface stated earlier can besimplified by means of the Gaussian curvature:

R(U, V)W = G - (a2(W) el - al (W) e2) dM2(U, V),

and, conversely, G can be expressed by the curvature tensor.

Theorem 19. The Gaussian curvature of a surface is

G = (R(e1, e2)e2, el) ,

where 1Z is the curvature tensor, and lei, e2} is an orthonormal frame ofvector fields.

Now we turn to the invariants of the second fundamental form, which weview, by means of the Riemannian metric, as a symmetric endomorphism

S : TM2 - TM2,

the Weingarten map of the surface introduced in the previous section. Itsnormalized trace is the mean curvature of the surface.

Definition 12. The mean curvature H of a surface is the function

H := tr (S)12.

The mean curvature of a surface can also be determined as the divergence(in R3) of the normal vector field.

Theorem 20. Let M2 be a surface, and let N be a vector field of length onedefined on an open neighborhood of M2 in R3. Assume that the restrictionNJnr2 of N to M2 is perpendicular to the surface. Then

H = 1 div(N),


where the divergence is taken with respect to the euclidean metric on R3.

Proof. From (N, dAl) = 0, we obtain

2H = (et, dN(e1)) + (e2, dN(e2)) + (N, dN(N))The last sum, however, is independent of the particular choice of the or-thonormal basis in IR3. This implies

2H = y'-, + ( ay j + (', Ar = div(N) . 0

Apart from the trace, the second fundamental form S has one more invariant,the determinant. But this does not lead to a new geometric quantity, as weshall now see.

Theorem 21. The determinant of the second fundamental form is theGaussian curvature,

det(S) = G.

Proof. Let x E M2 be an arbitrary point of the surface, and choose anorthonormal basis {e1, e2} consisting of eigenvectors of the symmetric endo-morphism in the tangent space TXM2. If 1C1i K2 are the eigenvalues, then,with respect to this basis, S has the matrix

rcl 00 K2

The assertion now follows directly from the Gauss equation,

G = (1.(e1,e2)e2,e1) = (II(e2,e2)S(el), e1) - (II(el,e2)S(e2), el)= Ic1 - 1£2 = det(S). 0

Remark. The preceding theorem allows to compute the Gaussian curva-ture, belonging to the inner geometry of a surface, from the second funda-mental form. Take an arbitrary (not necessarily orthonormal) basis v1i v2 inthe tangent space TXM2 at a point of the surface, and denote the matricesof the first and second fundamental forms by g and b, respectively,

__ 911 912 b11 b129

921 922 '9ij = I(v;,ZJ), b =

b21 b22J ,bij = II(v{,vj)

The Gaussian and the mean curvature can then be computed using theformulas

G det(g)and H = 2tr (b g-1) .

Their proof is a consequence of a more general observation. A symmetricbilinear form B on a real vector space V without any additional structurehas only a single invariant, the signature. If, however, a non-degeneratescalar product G is given in addition, then this induces an isomorphism


G : V' - V between the vector space V and its dual space V. Thus Bcan be viewed as a G-symmetric endomorphism, S := B o G-t : V -. Vin the vector space V, and all coefficients of its characteristic polynomialare invariants of B with respect to G. These, in turn, can be computed bymeans of an arbitrary basis, and the result does not depend on the particularbasis. Two of these invariants of the characteristic polynomial are the traceand the determinant.

Example 10. For the hyperbolic plane 7{2 from Example 9, a straight-forward computation shows that d&12 = a1 A a2. Therefore, the Gaussiancurvature is -1. A complete realization of H2 in 3-space does not exist(Hilbert, 1901). Certain open subsets of the hyperbolic plane, however, canbe realized as surfaces. The related mean curvature is not uniquely deter-mined; it depends on the particular realization as a surface in R3. One ofthem will be presented at the end of the discussion concerning surfaces ofrevolution to follow.

Example 11. For the surface of revolution from Example 3, we had

r"de3 = -z,,

4

This enables us to compute the Weingarten map explicitly:

S(a) i-j x' 88s 8s

S O'pJJ = z e2 = r 8V

Thus, in the basis 8/8s, 8/Ow, the Weingarten map is described by thematrix

r"/z' 0S =0 z'/r

From this, the Gaussian and the mean curvature are immediately computed:G=-r" H= 1 r"

r' 2

(z'r z')

Examples for surfaces of revolution with constant Gaussian curvature are thecylinder (C = 0, r = 1, z = a), the sphere (G = 1, r = sins, z = cos s), orthe pseudo-sphere (G = -1): The latter is an example of a (necessarily non-compact) surface of constant negative curvature, and hence a 2-dimensionalmodel of hyperbolic geometry. It is defined as the surface obtained as thesurface of revolution generated by the tractrix with the parametrization

(e_', ja 1 - e-2tdt I = (e-', -VI - e-2' + arctanh( 1 - e-28)) .


Example 12. The Gaussian curvature of the graph of the function h(x, y)(Example 4) is preferably computed using the differential of the form w12The resulting expression is

h== hyy - hiy(1 +h2

To compute the mean curvature, it is convenient to describe the graph asthe zero set of the function g : U -+ R, g(x, y, z) := h(x, y) - z. FromTheorem 5, §3.2, we then know that the normalized gradient vector field,

grad(g) 1

Y

(ham, hy, -1)1IIgrad(g)f J+ h.2 + h2

is perpendicular to the surface at each point. Hence the mean curvaturecan be computed as the divergence of this vector field (Theorem 20), andwe obtain

(1 + hy)hy= - 2h,hyhy + (1 + h2)hyyH =

2(1 + hZ + h2)3/2

Example 13. In order to apply this formula to the computation of theGaussian curvature for a modular surface (Example 5), we still have todetermine the second derivatives of h = If 1. From

h_ 8h+

8hand =_ Z (8h - 8h)

=ez e 8z 8z J

we first conclude formally that

_ a2h 82h 82hh=8z2

+ 28z8z + 0z2 '

.92h 82h _ 82h 102h - 82hhuy

_8z2

+2 8- 822' h,,,=

Z

8z2 8z2


After expanding, the numerator of the Gaussian curvature turns out to be

h.= hba - hz:y

We compute the derivatives:

_ (92h2 a2h 0h4 (idzoz) -

022 - a52 .

a2h fll.f (f1f)2 O2h f fN (fp)2 92h f, f,

az2 2 h 4 h3 ' azz 2 h 4 h3 ' azaz 4hfrom which we deduce the formula for the Gaussian curvature:

f,zG = (I+lf,12)2 Ref f"-1 .

In the example discussed before, f (z) = 1/ sin z, the Gaussian curvature hasthe form

G =

2Cost z + 1

12 [Re cost z- 1J

sill z 1+ cu52 z J

+ cos z

sine z

1212


Of particular interest is the curve on the surface separating the points of pos-itive Gaussian curvature from those of negative Gaussian curvature. Here,it is implicitly described by the equation

ReCost

z = 11 + cost z '

and is indicated as a dashed line in the figure on the previous page showingthe modular surface shaded according to the Gaussian curvature.

Example 14. We now continue the discussion of the surfaces defined by thereal and the imaginary part of a holomorphic function, f = it + iv : U C(Example 6). There we already derived from the Cauchy-Riemann equationsthe relation

l+uZ+uy = l+vi+vb = l+If'I2.A straightforward calculation yields the second derivative of f :

f" = uz2 - iu"V = v=y + iv==.

The formula for the Gaussian curvature of a graph thus leads to the remark-able fact that the surfaces defined by the real and the imaginary parts have


the same Gaussian curvature,

-I fui2G = (1 + I f,I2)2

The picture on the previous page illustrates this fact for the function f (z) _z2, again shaded according to the values of the Gaussian curvature. Thesurface becoming pointed at the comers belongs to Im z2, and the saddle-like surface corresponds to Re z2. The Gaussian curvature is

-4

(1 +4IzI2)2

On the other hand, the mean curvatures of these surfaces differ:

H(Re z2) = 4 (y2 - x2), H(Im z2) = -8 xy

(1 + 4x2 + 4y2)3/2 (1 + 4x2 + 4y2)3/2

Now we want to interpret the formula for the Gaussian curvature express-ing it as the determinant of the second fundamental form in a differential-geometric way. To this end, consider the normal vector e3 as a map on thesurface M2 with values in the unit sphere S2 C R3. The volume form dS2of the sphere evaluated on two vectors V, W E T=S2 at the point x E S2 isthe vector product

dS2(V.W) = (VxW,x).This formula is a special case of the general one for the volume form of thesphere S"'l C R" stated in §3.4, Example 28. From this, the 2-form e3(dS2)induced on M2 is easily computed:

e3(dS2)(V, W) = (de3(V) x de3(W), e3)

= ((w3l(V)el +w32(V)e2) x (w31(W)el +w32(W)e2),e3)= - w13 A w32(V, W) = -dwi2(V, W) = G dM2(V, W).

We summarize the result in

Theorem 22. Let e3 : M2 S2 be the unit normal map of the surface,and let G be its Gaussian curvature. Then the induced 2 -form e3(dS2) isequal to the volume form multiplied by the Gaussian curvature:

e3(dS2) = G - dM2.

Fix a point x E M2 on the surface and denote by D(x, E) the set of all pointson the surface whose distance from x is less than E. Then the two-dimen-sional surface area of the set D*(x, e) := e3(D(x, E)) C S2 can be computedusing the transformation formula for the integral as follows:

volS2 (D'(x,E)) =J

dS2 = ±J

e3(dS2) = ±J

G C. dM2.

D(x,e) D(x,e) D(x,e)


Dividing both sides by the surface area, volM2(D(x,e)), and applying themean value theorem of integral calculus for continuous functions, we obtainthe formula

IG(x)I = ovolss(D*(x,e))

£1 volA,2(D(x,e))

This was the original geometric definition of the curvature for a surface in 3-space. It is-similarly to the curvature of a curve-the infinitesimal volumedistortion of the normal map of the surface. For this approach to curvature,it is at first not obvious at all that the Gaussian curvature depends only onthe first fundamental form of the surface. This fact forms the contents ofthe Theorema Egregium (Gauss, 1827). Proceeding, however, like we didhere, this becomes immediate.

The total Gaussian curvature of a compact, oriented surface without bound-ary depends only on the topology of the manifold. We prove the so-calledGauss-Bonnet formula, expressing this fact explicitly. To do so, we need apreparation.

Lemma 3. Let V be a vector field of length one on the oriented surface M2,and let {el,e2} be an orthonormal frame in this orientation. RepresentingV as a linear combination, V = cos r el + sin r e2, the 1 form

Xv w12 + dr

is independent of the chosen frame. The exterior derivative of Xv is

dXV = - G dM2.

Proof. Every other frame el, e2 can be written as

el =e2 = -

and we already proved the formula w12 = w12 + dip. Moreover, r' = r - cp,and hence w12 + dr* = w12 + dr . 0

Stokes' theorem now immediately implies

Theorem 23. Let M2 be an oriented, compact manifold without boundary,and let V be a vector field of length one defined outside finitely many pointsx1, ... , xk E M2. If D(xi, E) C M2 are disk neighborhoods of the points xiof radius E > 0, then

r / k 1

J G dM2 = lim I J XvJ12


Hence we see that the integral of the Gaussian curvature is independentof the geometry of the surface. We evaluate the last formula explicitlyby choosing special vector fields. To this end, consider a smooth functionf : Aft -+ IR all of whose critical points are non-degenerate, and denote thenumber of its critical points with corresponding index by

rno (f) := the number of minima of f ,

nz z (f) the number of saddle points off,

M2(f) := the number of maxima off.

The vector field V = grad(f)/11grad(f) II is defined outside the set of criticalpoints. Computing the limits on the right-hand side of the formula, weobtain

2zr if x; is a minimum orlim J Xv = 5 -2a if x; is a saddle point.0 OD(z,.e) l

To prove this, choose coordinates x, y with respect to which the function fhas normal form, f (x, y) =

2(± x2 ± y2). We are only interested in the limit

of these integrals for a 0, and hence we may, without loss of generality,take the metric to be flat. From

gradf = f x a t y a11grad(f)II x2 + y2 ax 7X=2=+7y2 ay

we obtaincosy

xy

x2 -+y 2, sin r

VI_X2 -+y2

Computing the differential dr leads to the formula

dr = f x2 + y2

Here we have to take the positive sign in the cases of a minimum or amaximum, and the negative sign for a saddle point. Since

dr = ±21r,JJJSI

we obtain the limit we wanted to compute, and hence we have proved theGauss-Bonnet theorem.

Theorem 24 (Gauss-Bonnet Formula). Let M2 be a compact. oriented sur-face without boundary, and let f : M2 - R be a smooth function all of whosecritical points are non-degenerate. Then

ff2G dM2 = 21r (rno(f) - ml(f) + rn2(f)) .

maximum;

where 1110(f), MI M, m2(f) denote the number of minima. saddle points, andmaxima of f. respectively.


Corollary 1. Let M2 be a compact, oriented surface without boundary.

(1) The integral of the Gaussian curvature depends only on the topologyof the surface, but does not depend on its fundamental forms.

(2) The alternating sum mo(f) - MI (f) + M2 (f) is independent of thechoice of the function f all of whose critical points are non degen-erate.

Definition 13. The Euler characteristic X(M2) of a compact surface with-out boundary is the number

X(M12) := mo(f) - MI(f) + m2(f)Using this number, the Gauss-Bonnet formula for compact, oriented surfaceswithout boundary can be written as

2nz G - dM2 = X012)

nfExample15. The Euler characteristic of the sphere S2 is 2. In fact. the

restriction of the height function h(x, y, z) = z to S2 = {(x, y, z) E 1R3 :

x2 + y2 + z2 = 1) has one minimum, one maximum, and no saddle points.Thus

1i'

C d1112 = 227r a

for every surface M2 which is diffeomorphic (but not necessarily isometric)to S2.

Example 16. The Euler characteristic of the torus T2 is zero. It is notdifficult to find, e.g., on the torus of revolution (see §3.1), a function withone minimum, one maximum, and two saddle points. Thus

jAf2G d1112 = 0

for every surface which is diffeomorphic to the torus.

The Gauss-Bonnet formula in its general form comprises an additional re-lation for every vector field V of length one with finitely many singularities.First we note that the limit

lim Xve-o JD(X.'0

can also be computed, along the same lines as for gradient fields, for anarbitrary vector field. Start by choosing coordinates y', y2 around the pointxi E A12 such that this point corresponds to y' = 0, y2 = 0. Then representthe vector field V in these coordinates,

V = V''

+ V2y2.


B

C

As before, we may assume, without loss of generality, that the metric is flatin this neighborhood of the point, since we do not want to compute theintegral itself, but only its limit. Then (Vl)2+(V2)2 = 1, and hence cosy =V 1, sin T = V2. From this we obtain - sin r dr = dV 1, cos r dr = dV2,and thus the form Xv = dr is given by

Xv = dr =Therefore,

Jlim Xv = J (V1 dV2 - V2 dVl) := 2a Ind(V, xi).E-O (x i,e) 1

This number is called the index of the vector field V at the singular pointx; E M2. Notice that integrating V1 dV1 +V2 dV2 along S1 yields the sameresult. The geometric meaning of the index is illustrated in the figure above.The picture on the left shows some vector field (the circle is only drawn forgreater clarity and has no direct relation to the vector field itself). Startingat A, move around the circle once, and assign to each point on the circle thenormalized vector in S' pointing into the direction of the vector field at thatpoint. In the case shown in this figure, one full turn in the positive directionon the left corresponds to one full turn on the right, but in the oppositedirection. Hence the index of this vector field at the indicated point is -1.From the Gauss-Bonnet theorem, we obtain

Theorem 25 (Hopf-Poincard). Let M2 be a compact, oriented surface urith-out boundary, and let V be a vector field of length one defined outside{x1, ... , xk}. Then the sum

k

Ind(V, X,) = X(M2)i=1

is independent of the choice of the vector field.


A further integral formula deals with the integral of the mean curvature. Itgoes back to Steiner (1840) and Minkowski (1900), and plays an importantrole in the theory of ovaloids.

Theorem 26 (Minkowski-Steiner). Let M2 C 1R3 be a compact, orientedsurface without boundary, and let e3 be a unit normal field. Then

J f2H(x) dM2(x) = Jtz

(x,e3(x)) G(x) dM2(x) .

Proof. Consider the three functions p2(x) = (M(x), e;(x)) on the surfaceM2 and compute their differentials using the structural equations:

dpi = al +w12 P2+w13P3,dP2 = a2 + w21 P1 + w23 P3,

dP3 = W31 P1 + W32 P2

Note also that the 1-form

w := P1 W23 +P2 w31

does not depend on the choice of the tangent frame el, e2. Thus w is aglobally defined form on M2. We compute the differential of w, making useof the structural equations as well as the formula for the differentials dpi.This yields

Stokes' theorem now implies the Minkowski-Steiner integral formula.

The eigenvalues K1, K2 of the second fundamental form are called the princi-pal curvatures of the surface at the point. Their arithmetic mean is the meancurvature of the surface, whereas their product coincides with the Gaussiancurvature,

H = 2(i+K2), G =

The points of a surface are divided into several types, depending on the signsof their principal curvatures. A point of the surface is called

(1) elliptic if Kl K2 > 0 ;

(2) hyperbolic if K1 K2 < 0 ;

(3) parabolic if Kl K2 = 0 and '4 + K2 > 0 ;

(4) flat if K1 = K2 = 0 .

Umbilic points are the points of the surface for which the principal curvaturescoincide, Kl = K2. An umbilic point is either elliptic or flat.


Example 17. The line of points with vanishing Gaussian curvature indi-cated in the modular surface of the function f (z) = 1/ sin z (see p. 160)consists of parabolic points. For this reason, it is called the parabolic curve.By definition, it separates the hyperbolic from the elliptic points.

Theorem 27. If the surface M2 C R3 consists exclusively of umbilic points,it is part of a plane or a sphere.

Proof. By assumption, the Weingarten map is a multiple of the identity,S = x Id. The Codazzi equation,

0 = VS = V(K Id) = V(rc) Id,

implies that K is constant. In case K = 0, the normal vector e3 is alsoconstant. In fact, de3 = w31 el +w32 e2 = 0. Thus M2 is part of a plane. IfK # 0, we obtain from r.-Id = S = de3 the equation d(e3-KM) = 0.Hence e3 - KM is a constant vector a. Thus,

Ila+KMII2 = 1,

i. e., M2 is part of a sphere.

Theorem 28. A surface M2 C R3 with vanishing Gaussian curvature, G0, and constant mean curvature, H 0 0, is part of a cylinder.

Proof. The first principal curvature vanishes, r.1 = 0, and the second princi-pal curvature is constant, K2 0 0. In an orthonormal frame el i e2 consistingof eigenvectors of the second fundamental form, we obtain the formulas

W 13 = 0 and w23 = K2 a2.

Differentiate both equations:

0 = d1)13 = W12Aw23 = K2W12Aa2,0 = w21 A w13 = dw23 = Kra da2 .

Since K2 54 0, on the one hand, w12 is proportional to a2, and a2 is a closedform, dal = 0. But then dal = W12 A a2 = 0, so that altogether we obtain

dal = 0, dal = 0 and w12 = 0.The structural equations imply

de1 = 0, de2 = r-2'0`2-e3, de3 = K2 a2 e2 .

Hence e 1 is constant, and the surface M2 lies on a cylinder whose axis pointsin the direction of el.

Theorem 29. There exists no piece of a surface M2 C R3 satisfying G =-1 andH =0.


Proof. Let r£ 1i rc2 be the principal curvatures. Then 0 H =2

(rcl + K2)and -1 = G = rclrc2 immediately imply rcl = -k2 = 1. Thus we obtain,with respect to a tangent orthonormal frame consisting of eigenvectors ofthe second fundamental form, the equations

w13 = al and W23 = - 02 .

Differentiating these equations yields

dal = d013 = w12 A w23 = - w12 A a2 ,

dal = - dw23 = - W21 A w13 = w12 A al .

On the other hand, the form w12 = A al + B a2 is a linear combination ofal, 0`2 with coefficients A and B still to be determined from

dal = A al A a2 and dal = B al A a2.But then

A al A a2 = dal = -w12 A a2 = -A al A a2,B al A a2 = dal = w12 A o1 = -B a1 A a2,

and hence A = B = 0. The form w12 = 0 vanishes, contradicting G =-1. 0Definition 14. A surface M2 c ]R3 with vanishing mean curvature, H =_ 0,is called a minimal surface.

The preceding theorem states that there does not exist any minimal surfacewith constant, non-vanishing Gaussian curvature. Consider, more generally,the normal map e3 : lbi2 - S2 of a minimal surface. From i1 = -rc2 =v G, we obtain the formulas

W13 = Gal, w23 = -V-v02.The relation de3 = w31 el+w32 e2 evaluated on tangent vectors V, W E TM2immediately implies

(de3(V), de3(W)) G (V, W).

If and IM2 denote the fundamental forms of the sphere S2 and the surfaceM2, respectively, then the induced fundamental form (Riemannian metric)e3(IS2) is proportional to IM2:

e3(IS2) _ - G IM2 .

Thus we have proved the following theorem.

Theorem 30. The normal map e3 : M2 -* S12 of a minimal surface pre-serves angles, and the Riemannian metric g' := - G Ir112 defined on M2 isa metric of constant positive Gaussian curvature.


The vector-valued position function M : M2 -R 3 of a minimal surfacegives rise to harmonic functions.

Theorem 31. Let M : M2 --+ 1R3 be a surface, and let 0 be its Laplacianacting on functions. Then M2 is a minimal surface if and only if the vector-valued function M is harmonic, 0(141) = 0.

Proof. The Laplacian can be expressed by the exterior derivative and theHodge operator (see §3.11). Using the structural equations, we compute asfollows:

O(M) dM2 = d * d(M) = d * (al el + a2 e2) = d (a2 el - ai e2)

= del Aa2+do2 el -de2Aal -dal e2= Aal - el

-(w21 - el -w12Aa2-e2

=

Corollary 2. There does not exist a compact minimal surface M2 C R3without boundary.

Proof. Otherwise, by Hopf's theorem (see §3.8), the components of theposition function would have to be constant, a contradiction.

The normal map e3 : 1412 -+ S2 of a minimal surface with non-vanishingGaussian curvature is a local conformal diffeomorphism. The stereographicprojection from S2\{north pole} to 1R2 is a conformal diffeomorphism, too.Hence, by inverting the normal map e3 : M2 -+ S2 IIt2, every minimalsurface can locally be parametrized by a map F : U -+ M2 such that

(*) OF, 49F = /8F 8F\ \8F OF)ayl , ayl J àye

OF,aye J

and aylOF, aye = 0.

Coordinates like these are generally called isothermic coordinates. The min-imality of the surface expressed in these coordinates is equivalent to therequirement that F = F(yl, y2) is harmonic with respect to the LaplacianA = 82/(ayl)2 + O2/(oy2)2 on R2,

A(F) = 0.

Identifying R2 with the complex numbers C, every harmonic map F : U -+1R3 can locally be represented as the real part of a holomorphic map 4U -+ C3; this is a direct consequence of the Cauchy-Riemann equations andPoincare's lemma. The real partial derivatives of 4) can be expressed by the


complex derivative (z = yl + iy2), since 4' is holomorphic:a4; a4

anda4 = ia4

ayl = az ay2 azDenoting the derivative 84'/az by W and writing 4< _ ('I'i,'1'2,'P3) withrespect to the coordinates in C3, condition (*) is equivalent to the quadraticequation

'I,j+*2+q13 = 0.The solutions of this equation are described by two arbitrary holomorphicfunctions f (z) and g(z):

4'1(x) = f z) (1 2(1 + 92(x)).- 92(z)), 'I2(z) = i f 13x= f(z)9(z)Summarizing, we obtain

Theorem 32 (Weierstrass Representation of a Minimal Surface). Everyminimal surface M2 C 1R3 with non-vanishing Gaussian curvature can locallybe represented in terms of two holomorphic functions f (z) and g(z). Aparticular parametrization is provided by the formula

F = Re(J f 2z) (1 - g2(z))dz, i J L_ z) (1 +92(z))dz, f f (z)9(z)dz).

Remark. A simply connected minimal surface always occurs as one min-imal surface within an S'-parameter family; it can thus be deformed. Infact, if 4i := 4'R + i 4i1 : U C3 is holomorphic, then the equation

Re(e7Q4i) = COs(a) 4iR - sin(s) 4i1

defines a family of minimal surfaces, since the quadratic equation 4i

+ %F2 +'p3 = 0 is homogeneous.

Example 18. Very nice non-trivial minimal surfaces already arise by insert-ing relatively simple functions. Consider, for example, the following tripleof holomorphic functions:

1-Z2 1+z2 1

2iz2 '2z2 , az)

satisfying the condition 'I' +'y2 +'I'3 = 0, and their complex primitives,

F = (Ft, F2, F3) = (_(z + 1/z),2

(z - 1/z), In z)

The real and the imaginary parts of these functions determine a pair ofminimal surfaces which are called adjoint to one another. Setting In z =g + ir, these are described by the formulas

FR := Re(F) _ (- sinh a sin r, sinh a cos r, r),Fj := Im (F) _ (- cosh p cos r, - cosh p sin r, q) .


Hence FQ = cos a FR+sin a F1 determines a 1-parameter family of minimalsurfaces joining FR to Fl. The surface FR is a helicoid, whereas F, is thesurface of revolution generated by the catenary 2, the catenoid. The sequenceof pictures above illustrates the deformation of both these minimal surfacesinto one another. Further examples of classical minimal surfaces can befound in the exercises.

5.5. Curves on Surfaces and Geodesic Lines

Consider an oriented surface M2 C R3 in 3-space, and on it a curve 7 :[a, b] M2, which we assume to be naturally parametrized. The tangentvector t(s) to the curve is also tangent to the surface, and will be denotedby el(s). If e3(s) is the normal vector to the surface AI2 along the curve,the equation el(s) x e2(s) = e3(s) determines a third vector, which is againtangential to M2 along the curve. This way, we construct at each point ofthe curve an orthonormal frame, the so-called Darboux frame of the curve

2We recall that this is the curve formed by a perfectly flexible inextensible chain hangingfrom two supports.

5.5. Curves on Surfaces and Geodesic Lines 173

'y with respect to the surface M2. Unlike the Frenet frame, the Darbouxframe of a curve in a surface is defined at every point of the curve, even atthose points where its curvature vanishes. Now we decompose the curvaturevector k(s) := K(s) h(s) of the curve y in R3 into the part kg(s) which istangential and the part kn(s) which is perpendicular to the surface.

Definition 15. The vector field kg(s) is called the geodesic curvature vector,the vector field kn(s) the normal curvature vector of the curve with respectto the surface,

k(s) = kg(s) +

The vector icn(s) e3 is proportional to the normal vector ofthe surface with a factor Kn(s), called the normal curvature. Analogously,kg(s) := -Kg(s) e2 is proportional to e2, since the principal normal vectorh(s) is orthogonal to the tangent vector. From this we obtain the geodesiccurvature Kg(s) of the curve, and the square of the curvature K(s) of thecurve is the sum

K2(S) = K92 (S) + K2(S).

Rewriting the geodesic curvature,

Kg(S) = (kg, e2) = (k, e2) = -(del(el), e2) = -I(Veiel,e2),we are led to the following theorem.

Theorem 33. The geodesic curvature of a curve in a surface depends onlyon the first fundamental form of the surface. In the Darboux frame.

K9(s) = - I(Velel, e2)

Similarly, the normal curvature Kn(s) of a curve can be expressed by thesecond fundamental form of the surface and its tangent vector:

K. = (kn, e3) e3) = (e3ds /

_ (de1(eI), e3)

Theorem 34. If y and y` are any two curves tangent at a point of thesurface M2, their normal curvatures at this point coincide.

Now we consider a variation of the curve y : [a, b] 1412 within the surface1112 with fixed initial and end points. We may imagine this as a family ofcurves,

yE (s) := y(s) + e h(s) e2(s) + O(e2),determined by a function h vanishing at the ends of the interval, h(a) _h(b) = 0. The length of the curve yE is

fbIIs)+ed(s).e2(s)+E.h(s).(s)+0(e2)II,L((s)) =

174 5. Curves and Surfaces in Euclideai 3-Space

and computing the derivative with respect to the parameter e yields

=jb

(s)e2(s) + h(s) d 2 (s)) ds .

The formulas (t(s), ez(s)) = 0 and

(t(s), p(s)) = (e1, ve,ez) = w21(el) = ng(s)

then, eventually, imply

de (L(7£(s))) Lo

b

= f h(s) sg(s)ds.a

Theorem 35. Let 7 : [a, b] - M2 be a curve in the surface, realizing theshortest distance between the points y(a) and 7(b) within the surface. Thenthe geodesic curvature of the curve vanishes,

r..9(s) = 0.

Definition 16. A curve in the surface is called a geodesic line if its geodesiccurvature vanishes. This condition is equivalent to VF= 0.

If the geodesic curvature Kg = 0 vanishes, the curvature vector

ac(s) h(s) = k(s) = k' (s)is proportional to the normal vector e3 to the surface. Hence a curve withnon-vanishing curvature, K(s) # 0, is a geodesic line in the surface.bfz if andonly if the principal normal vector h(s) coincides with the normal vector e3to the surface,

h(s) = ±e3(7(s))Geodesic lines realize the shortest distance between two points of the surfaceonly if the points lie sufficiently close to one another. A longer geodesic linedoes not necessarily have to be the shortest path between its end pointswithin the surface. Examples-on S2 or on the cylinder-are easily con-structed (see Example 21). We represent the curve in local coordinatesy1,y2 on the surface as y(s) = (y'(s),yz(s)). If g3(y',yz) are the coeffi-cients of the Riemannian metric and s is the natural parameter of the curve,then

2 ;dp'1: gti(y'(s),yz(s))

dy

ds ds = 1.i j=1

The Christoffel symbols r occur in the representation of the covariantderivatives of the basis vector fields 8/8y` as linear combinations of them-selves:

z

ey y7 Eraykk=1

5.5. Curves on Surfaces and Geodesic Lines

Since

175

8 2dyi k 8ds - ayk ,

i.k=1

the following formula holds for the tangent vector t :2 k i°t{

- ds2 +I

ds ds aykk=1 1.3=1

We summarize this observation in the following theorem:

Theorem 36. Let -y (s) = (y1(s), y2(s)) be a curve represented in local coor-dinates. It is a geodesic line if and only if the following system of differentialequations is satisfied (1¢ = 1, 2):

d2yk 2

ds2+ E

i j=1rjasa =o.

This is a second order system of differential equations. The general existenceand uniqueness result from the theory of ordinary differential equations en-sures the existence of precisely one geodesic line through any fixed point ina given direction.

Corollary 3. For each point x E M2 and every tangent vector V E T,M2 oflength one on a surface without boundary, there exists exactly one geodesicline -y : (-e, e) -+ M2 satisfying the initial conditions y(0) = x, Y(0) = V.

Example 19. Let -y(s) be a geodesic line on the two-dimensional sphereS2 of radius R > 0. Since x9 = 0, the normal curvature sn and the spatialcurvature a of the curve coincide. Hence

= Ian l = II(t, t) = R

since the sphere consists exclusively of umbilic points. The normal vectorto the sphere is e3(x) = 11 x. Thus, on the one hand, we obtain

ds2) s(k(s)) = ds(lcn(s)) = -R ds(e3('y(s))1 dry (s) 1

R2 ds = - R2 r(s)and, on the other hand, the Frenet formulas for the curve imply

d2t(s) ds)

t(s) + r b(s) .Rds2 R ds R2

Consequently, the torsion r of the curve vanishes identically. The curve -y(s)is thus a plane curve in S2 with constant curvature 1/R, i.e., a principalcircle in S2.

1


Example 20. The relation K 2 = n9 + I£,2a immediately implies that everycurve with vanishing curvature is a geodesic line. Formulated differently, ifa straight line of R3 lies in a surface, then it is necessarily a geodesic line inthis surface.

Example 21. Consider the cylinder M2 in 1R3 whose axis is parallel to theunit vector a5. The normal vector at a point x E M2 is proportional to theprojection of x onto the plane perpendicular to a,

e3(x) = x - (x, a)a

lix - (x, a)611

For a geodesic line -y(s) on the cylinder, we immediately obtain from h&(s) =e3(y(s)) the equations (h(s), = 0 and

C := (h(s), 'Y(s)) = (e3('Y(s)), 'Y(s)) = hh'Y(s) - ('Y(s), a)d1l

Here C is positive and constant, since y(s) lies on the cylinder. As dt(s)/ds =K(s) hs(s) and db(s)/ds = -r(s) hs(s), the equation (h(s), a) = 0 impliesthat

A := (t(s), a) and B := (b(s), a-)are constant. In case A = 0, the curve lies in an affine plane perpendicularto d and on the cylinder; hence it is a circle. For A 34 0, we differentiatethe equation (hh(s), a) = 0 once, and then the Frenet formulas imply therelation

-A Ic(s) + B r(s) = 0.Differentiating the constant C2 twice, we deduce that

C ic(s) + 1 = A2.The curvature K(s) as well as the torsion r(s) are thus constant, and hencethe curve y(s) is a helix (see Exercise 6). In summary, the geodesic lines onthe cylinder are helices, circles, and the straight lines parallel to the 5-axis.Two points on the cylinder may be joined by infinitely many geodesic lines,only one of which is the shortest path between them:


The equations characterizing geodesics are non-linear differential equationsof second order, and hence, in general, difficult to solve. Things become a lit-tle easier if, because of the symmetry of the surface, one can find sufficientlymany functions which are constant along geodesics:

Definition 17. A function f : TM -+ llt is called a first integral of thegeodesic flow if f is constant on all tangent curves y'(t) of geodesics.

The length of y' itself is always one non-trivial first integral, which is inter-preted as energy. It expresses nothing but the fact that geodesic lines areparametrized by arc length. Further first integrals are obtained by Noether'stheorem starting from isometries of the surface.

Theorem 37 (Noether's Theorem). Let V be a tangent vector field on thesurface M2 whose flow 4bt : M2 -+ M2 consists only of isometries. Then thefunction

fv : TM -i R, fv(W) = l(W, V),is a first integral of the geodesic flow.

Proof. Consider a geodesic line y(s) on M2 in its natural parametrization.Setting W = y', the function defined in the theorem has the following de-rivative with respect to s:

ds (Y'(s), V(7 (s)) _ (7'(s), V(Y'(s)) + y'(s), dV(y'(s)))

The vector -y" is nothing but the curvature vector of the curve in R3, andfor a geodesic line this is always perpendicular to the surface. Hence thefirst summand vanishes. On the other hand, the flow Ot was supposed toconsist of isometries,

(W1, W2) = (d't(Wi), d0t(W2))

Differentiating this identity and evaluating the result at t = 0, we obtain

(dV(W1), W2) + (dV(W2), WI) = 0,

and the second term vanishes, too, since it has to be antisymmetric in y'(s).O

Example 22. We want to find all geodesic lines on a surface of revolution.For simplicity, assume as before that the generating curve is given in naturalparametrization. With respect to the coordinates s and V on the surface, ageodesic line is parametrized as y(t) _ (s(t), ap(t)) and has the first integral

E := I1-r'(t)II2 = s 2 + r(s)2V2


Since the rotation through the angle V around the z-axis is an isometry ofthe surface, the vector field 8/8W satisfies the assumptions of the Noethertheorem and gives rise to a further first integral,

M := (y', 8/8V) = r2c .

The existence of this second invariant is known as Clairaut's theorem (1731)and has the following geometric interpretation. Consider a geodesic linethrough a point of the surface, and also the meridian 77(t) = (so, W(t))through this same point. The angle a formed by the geodesic and themeridian is computed as

_ ('y', W) r2Vt2 r Icos(a)

11Y11 ' 11741 E r2Vp' ETherefore, M is constant if and only if r(s) cos(a) is constant. For thequalitative discussion of the geodesic lines, we take E = 1. Inserting yY _M/r2 into E = 1, we obtain

s 2r2 = r2 - M2, thus 0 < M2 < r2 .

If M vanishes identically, then gyp' = 0, and p has to be constant. Thiscorresponds precisely to the generating curve of the surface of revolution;each profile curve of F is thus a geodesic. In the other extremal case,M2 = r2, we have s' = 0, i. e., a has to be constant. This curve is ameridian on F; note, however, that in this case differentiating M2 = r2immediately leads to r r' = 0, which, in turn, implies r' = 0. A meridianof F is a geodesic if and only if the radius function r has an extremal pointthere. All that remains is the generic case, 0 < M2 < r2. The conditionE = 1 is equivalent to

a' = fl r2-M2.rIn order to eliminate the curve parameter t, we view s as a function of gyp;hence ds/dcp = s'/V,

= fM r2 - M2,d,p

and thusrSOP) 1 r,v

MJ ds=J do =fop.,o r(s) r(s) - M2 0

This provides a parametrization of the geodesic lines in the case of surfacesfor which the integral on the left-hand side is elementarily integrable andcan be solved for s.


Example 23. Let 7.(2 be the hyperbolic plane (see Example 9). We againparametrize the geodesic ry as 'y(s) = (x(s), y(s)). Then the condition that-1 is parametrized by arc length takes the form

xi2 i2

y2 + y2 = 1 .

Since the metric does not depend on x, translations in the x-direction areisometries; the associated vector field is 8/8x. Hence

M :_ (y', 3/ex) = lye

is a first integral. If M vanishes, x has to be constant, and the geodesiclines are precisely the open straight half-lines parallel to the y-axis. In thegeneral case, rewrite the first condition, inserting M. A short calculationleads to

y = ±y l - M2y2.Again, we view y as a function of x and set r:= 1/jMI,

dy = y 1 r2 _ y2dx x' y

Directly integrating this equation and denoting the integration constant bya yields

- r2 - y2 = (x - a) hence (x - a)2 + y2 = r2.

The trace of the geodesic line is thus a semi-circle whose center lies on thex-axis.

Once again we now turn to the Gauss-Bonnet formula. Let M2 be an ori-ented, compact surface with boundary 9M2, and let V be a vector field oflength one defined on M2 except at finitely many points xi, ... , xk in theinterior of M2. Assume, moreover, that V is perpendicular to 9M2 at theboundary 3M2 and outwardly directed. Consider the 1-form Xv introducedin §5.4. From the formulas proved there, we deduce

k

J dXv = J

aM2

Xv - 27r Ind (V, xi).

Near the boundary 8M2, we choose an orthonormal frame el, e2 such thatel = V. Then Xv = w12, and e2 is the unit tangent vector to the boundary.Since W12(e2) = I(Dezel, e2) = -I(el, V2e2) = K9, we obtain the Gauss-Bonnet formula for manifolds with boundary.

Theorem 38 (Gauss-Bonnet Formula). Let M2 be a compact, oriented sur-face with boundary, and let V be a vector field of length one defined on M2except at finitely many points xl, ... , xk in the interior of M2. Assume,


moreover, that at the boundary V is perpendicular to the boundary and out-wardly directed. Then

kf fJ G dM2 + J icy d(8M2) = 2a Ind(V, xi).

,112 0M2 i=1

This has similar consequences as in the case of surfaces without boundary.On the one hand, the sum of the integrals

IMZ8

f1112 K9.d(aM2)

is independent of the first fundamental form. On the other hand, the sumof the indices of a vector field does not depend on the vector field.

5.6. Maps between Surfaces

The length IIVII = I(V, V) of a tangent vector can be determined bymeasuring the length of curves. To see this, choose a curve -y on the surfaceJ112 in arbitrary parametrization whose tangent at t = 0 coincides with V,y(O) = V. The length of the curve segment y([0, t]) is defined as

L(t) = f I("Y(Ft),'Y(p))dp ,0

and computing the derivative at t = 0 yieldsdL(t)

I(V, V) .dt t=o

The first fundamental form of the surface is thus completely determined bythe lengths of curve segments, and vice versa. Hence each map f : M2 M2between two surfaces which does not alter the lengths of curve segments hasa differential df : TM2 -+ TM2 preserving the first fundamental forms I,1f2and IAt2,

I (IT12) = IJ12.

Definition 18. An isometry f : M2 M2 between two surfaces is alength-preserving smooth map.

The Gaussian curvature of a surface is a geometric quantity depending onthe first fundamental form. This immediately implies

Theorem 39. Let f : M2 -+ M2 be an isometry. Then the Gaussiancurvatures G,2, Gh12 at corresponding points coincide:

GM2(x) = Gh12(f(x))Corollary 4. There does not exist an isometric map from an open subsetof S2 onto an open subset of the euclidean space R2.

5.6. Maps between Surfaces 181

For this reason, cartography was forced to look for other suitable maps. Es-pecially important are conformal (angle-preserving) and volume-preservingmaps. Let us first turn to the former ones. A map f : M2 AI2 is calledconformal (or angle-preserving) if the angle between two intersecting curves11,12 in M2 is the same as that of the image curves f o 11, f o y2 in R12.

These maps can be characterized using the first fundamental form.

Theorem 40. A map f : M2 11 I2 is conformal if and only if there existsa positive function h : M2 - (0, oc) such that f `(ITf2) = h I tif2.

Proof. This is an easily proved fact from linear algebra. Let ( , ) 1 and 2

be two positive definite scalar products on a two-dimensional vector spacesatisfying the equation

(V, W)1 (V, W)2IIVII11IWII1 IIVII2IIWI12

for all vectors V. W. Then there exists a positive number h such that

(V, W)2 = h (V, W)1 .

We already encountered conformal maps. The normal map e3 : M2 _ S2of a minimal surface in R3 is conformal (Theorem 30). Complex analyticfunctions are conformal maps from the plane R2 to itself. In fact, if f =(u, v) = u + i v is a map f : 1R2 1R2, then the following formula holds forthe first fundamental form IR2 = dx2 + dy2:

f'(Ia2) = (utdx+ul,dy)2+(v.dx+vydy)2= (u2 + v2)dx2 + 2(uruy + v=vy)dx O dy + (v2 + vy)dy2.

The Cauchy-Riemann equations, ur = vy, uy = -vi, imply that, for theinduced form,

f(1R2) = If,(z)12.Ia2.

Hence every complex-differentiable function f = u + iv is a conformal map,and the square of the absolute value of its derivative, I f'(x)12, appears as theconformal factor in the sense of two-dimensional geometry. The best-knownconformal map is the stereographic projection from the two-dimensionalsphere to the plane. This shows that conformal maps from R2 to S2 ex-ist, and so angle-preserving geographic maps can be drawn.

Example 24. In coordinates, the stereographic projection f : R2 S2 isthe map

2x 2y 1 - x2 - y2(l+x2+y2' 1+x2+y2' 1+x2+y2).


The partial derivative vectors of f are

Of 2 (1 - x2 + y2, -2xy, -2x) ,Ox - (1+x2+y2)2Of

-2

(I + x2 + y2)2 (-2xy, 1 + x2 - y2, -2y)

and hence

Of Of _ 4 _ Of Of 8f Of\ 8x' 8x (1 + x2 + y2)2 8y' ay

and8x' 8y _ 0.

These formulas show that the stereographic projection is conformal.

Example 25 (Mercator projection). Let (gyp, tai) be spherical coordinateson the sphere S2\{N, S} outside the north and the south pole (see §3.2,Example 14). Define a map from the sphere S2 to R2 by the formula

(V, log (tan(4 + 2111This turns out to be a conformal map with an additional interesting prop-erty: curves of constant azimuth (i.e. forming constant angles with themeridians on S2) are mapped to straight lines under f. For this reason, theMercator projection (1569) has been one of the most widely used methodsfor the creation of nautical maps for centuries. The reader may find anextensive discussion of further important map projections in the book byK. Strubecker (Volume 2) cited in the bibliography.

Another interesting class of maps between surfaces are those preserving thetwo-dimensional volume (surface area) of measurable subsets of the surface.Such a map f : M2 --+ M2 is characterized by the property that the volumeform is preserved,

f*(dM2) = dM2.

We will call such an f an area-preserving map. Isometries are the mapswhich are both conformal and area preserving.

Theorem 41. A map f : M2 M2 is an isometry if and only if f isconformal and area-preserving.

Proof. The relation f*(IM2) = h. IM2 implies f*(dM2) = h2d.112' andhence also the assertion.

The sphere S2 (or pieces of it) thus cannot be mapped to the plane inan angle- and area-preserving way. There exist, however, angle-preservingmaps, as the stereographic projection shows. We now want to describe anexample for an area-preserving map from S2 to 1R2.

5.7. Higher-Dimensional Riemannian Manifolds 183

Example 26. Delete the north and the south pole from the sphere S2, jointhem by a straight line, and consider the cylinder whose axis this line is:

Z = {(x,y,z) E R3: x2+y2 = 1}.To each point different from the north and the south pole, P 0 N, S, therecorresponds one straight line containing the point P which is perpendicularto the fixed axis. This line intersects the cylinder in two points, and wedenote the one closer to P by f (P). This defines the so-called Lambertprojection,

f : S2\{N,S} -> Z.Composing this with an isometry from the cylinder to the plane, we obtainan area-preserving map from S2 to R2.

To see this, let el, e2 and ei, e2 be the orthonormal frames on S2 and Z.respectively, indicated in the above picture. Then an elementary geometricargument immediately leads to the relation

df(ei) = rei, df(e2) = Tee

Here r is the radius of the meridian of S2 tangent to e2. These formulasprove that f is an area-preserving map.

5.7. Higher-Dimensional Riemannian Manifolds

Following §3.11, we will conclude this chapter by a brief survey of higher-dimensional Riemannian manifolds. Only the basic facts are discussed, andno comprehensive introduction into Riemannian geometry is intended. Asin the case of surfaces, every pseudo-Riemannian manifold (11I'", g) can beendowed with a covariant derivative V of vector fields with respect to vector


fields which is uniquely determined by two conditions. This formalizes thenotion of parallel displacement of vectors on a manifold. In fact, Levi-Civita's idea (1916) that every Riemannian space admits a unique paralleldisplacement preserving the metric turned out to be fundamental for thefurther development of geometry.

Theorem 42. Every pseudo-Riemannian manifold has precisely one covari-ant derivative V W of vector fields with the following properties:

(1) V (W1 + W2) = V (Wi) + V (W2)

(2) Vv, +v2 W = Vv, W + Vv, W ;

(3) if f is a smooth function, then

Vv(f-W) = and V f.vW = f VvW ;

(4) Vg(WI, W2) = g(VvW1, W2) +g(W1, VvW2)

(5) VvW - VwV = [V, W).

The vector field VvW is determined by the expression

2g(VuV, W) = U(g(V,W))+V(g(U,W))-W(g(U,V))+g([U,V}, W)

+g(V, [W, 141) - g(U, [V, W])

Proof. We prove uniqueness of the covariant derivative and derive in par-ticular the stated formula for VvW. By assumption, we have

U(g(v, W)) = g(VuV, W) + g(V, V W) ,

V(g(U, W)) = g(VvU, W) + g(U, V W),W(g(U, V)) = g(VwU, V) + g(U, V WV).

Because of the properties of V, we may introduce the commutators of thevector fields into these equations:

W(g(U,V)) = g(VuW,V)+g(U,VvW)+g([W,U],V)-g(U, [V, W]),V(g(U,W)) = g(VuV,W)+g(U, VvW)-g(W, [U,VI)

From the first, fourth, and fifth equation we obtain the formula for the scalarproduct g(VuV, W). Hence there exists at most one covariant derivativewith the required properties. Conversely, the formula determines the vectorVvW, and an analogous computation shows that the operation thus definedis a covariant derivative with all the properties needed.

Definition 19. The covariant derivative V W is called the Levi-Civitaconnection of the pseudo-Riemannian manifold.


Let V1, .. . , Vm be an arbitrary frame of vector fields defined on an opensubset of Mm. Representing the covariant derivative of the basis vectorfields as

EU]

OV,Vj 17 - Vk,13

k=1

we introduce the Christoffel symbols of the Levi-Civita connection. Thesymmetry properties of the functions r strongly depend on the chosenframe. If the vector fields Vi mutually commute, [Vi, Vj] = 0, and if gij =g(Vi, Vj) are the coefficients of the Riemannian metric, then formula (5)from Theorem 42 simplifies, and we obtain the expression

m

rj = 9>9'(Vi(9j0)+Vj(9ta)-Va(9ij)).

a=1

In this case, the Christoffel symbols are symmetric in the lower indices,

r = r (for commuting frames).

If the frame V1 := el, ..., Vm := em consists of mutually orthogonal vectorfields of constant length ±1, then the equality

0 = Vi(g(Vj,Vk)) = 9(VV,Vj,Vk)+9(Vj,VV.Vk)

implies the following symmetry property of the Christoffel symbols:

r = -rk (for orthonormal frames).

Example 27. The first fundamental form of a surface of revolution can bewritten (see Example 3) as

I = ds2 + r(s)2 dcp2 .

With respect to the commuting frame V1 = 8/8s, V2 = 8/8cp we obtain, forthe "classical" Christoffel symbol rte,

I'l2r g

22 , IaQ p9 + a-g ,0 - as9"''J 2 [

8as )2 J

-rr '.;=8

and similarly rig = r21 = r'/r. The other Christoffel symbols vanish.From Example 3 we take the formula W12 = (r'/r) 02 with respect to theorthonormal frame el = 8/8s, e2 = (1/r) 8/8cp and its dual frame ?1,02considered there. Together with the definition of the "Cartan" Christoffelsymbols,

W12 = r11 ai + r21 o2 = 0.01 + -a2,

this yields r2 1l = 0 and r2, = r'/r. Similarly we have

W21 = r11 o1+r1 a2 =12 22 2 r


implying r12 = 0, r22 = -r21.

Again denoting by Q1, the 1-forms dual to the frame e1.... , c,,,, weintroduce the local connection forms

wij := Ej -9(Vei,ej) = oj(Vei) with Ej :=g(ej,ej)By definition, they satisfy

m

wij = rQi - va and Ei - wij + Ej wji = 0 .a=1

N%,e compute the exterior derivative doi using the properties of the Levi-Civita connection:

dui(V,W) = V(Qi(W)) - W(oi(V)) -oi([V,WI)M

_ Eej-{9(Vvei,ea).a.(W)-9(Vwei,ea)-a.(V)}a=1M

_ 1: aQAw,,i(V,W)a=I

This leads to the first structural equation of a pseudo-Riemannian space.In order to formulate the second structural equation, denote by SZ := (wi j )the antisymmetric (m x m) matrix composed of the 1-forms 4;ij. This addi-tional structural equation computes the matrix d!2 - S1 A S2 of 2-forms, andto formulate it we need the curvature of the space. The curvature tensor7 (U. V)W := VuVVW-VVOuW-V1U,VIW of a pseudo-Riemannian man-ifold is introduced generalizing the case of surfaces, and all the identities ofTheorem 15 in §5.3 remain valid. In particular, the first Bianchi identityholds:

R(u, V)W + R(V,W)u + R(W,u)V = 0.However, in the higher-dimensional case the curvature tensor does not re-duce to a single function. By definition, it is a (3,1)-tensor. This can betransformed into a (4, 0)-tensor, the Riemannian curvature tensor:

R(U,V, W, Wi) := 9(R(U, V)W, W1) .Referring to an orthonormal frame e1, ... , e,,, of basis vector fields of lengths±1, we obtain from

m

R(ei, ej )ek Rijk el1=1

the components Rijk of the curvature tensor. These can be computed bythe formula

Rtijk = El . R(ei, ej, ek, el)

from the Riemannian curvature tensor.


Theorem 43 (Structural Equations for Pseudo-Riemannian Manifolds).Let el, ... , em be a local orthonormal frame on a pseudo-Riernannian man-ifold, and let o1, ... , an be the dual frame. Then the following equationshold:

Lul

(1) dai = 1: a. A wai ;Q=1m 1 m

(2) dwij = win A waj +2

as A a3 .a=1 a,J3=1

Proof. We already derived the first equation. To prove the second, weuse the properties of the Levi-Civita connection and obtain, after severalelementary steps,

dwij(V,W) = a (R(V, W)ei) +Ej - (g(Vwe;, Vvej) - g(Vvei, Vwej))71'

_ aj(R(V, W)ei) + ejEa(Qa(Vwei)Qa(Ovej) - aa(Vwej)an(Vvei))a=1

Moreover,

ej - e -an(Vvej) = -wnj(V)

and, similarly, ej en aa(Vwej) _ (W). Altogether, this completesthe proof,

m

dwij(V,W) _ winAwaj(V.W)+oj(R.(V.W)ei)a=1

_ wiaA waj(V,W)+RVWi 0a=1

Corollary 5. If the curvature tensor of a pseudo-Riemannian manifold X11"'vanishes, then in a neighborhood of each point there exists a chart in whichthe coefficients of the Riemannian metric are constant,

g = diag(±1, ....±1).

Proof. First choose a local orthonormal frame e1, ... e.. and consider thematrix n = (wij) of connection forms. Then

dfl = SZ A Q,

since the curvature tensor vanishes. By Theorem 6 in §4.3. there exists a

(pseudo-)orthogonal matrix A of functions such that

ft = dA A-1 and A(O) = diag(± 1.... , ±1).


Denote by r := (a1 ... a,,,) A a special system of 1-forms determined bya1, . . . , o,,, and the matrix A. The structural equations imply that this formis closed.

dr = 0.

Hence, by Poincare's lemma, each 1-form r, can locally be represented as thedifferential of a function fi. This way, we construct local coordinates on themanifold, and for the pseudo-Riemannian metric g we obtain the expression

g = ±ai + ... t a2, = ±rl + ... f T'2 = ±(dfl )Z + ... f (dfm)Z.

Definition 20. A pseudo-Riemannian manifold with vanishing curvature'tensor is called flat.

The preceding corollary thus states that every flat pseudo-Rieinannian spaceis locally isometric to the standard space Rm.k with the following metric ofindex k:

dxl &I +... + dxm-k dXin-k - dxm-k+l , dXm-k+l - ... - dx"' ) dx'n.

Consider a curve -y(t) in the pseudo-Riemannian manifold (111',g) and avector field W(t) along this curve. In addition, we may suppose that locallythe latter is the restriction of a vector field W defined on Mm. In the neigh-borhood of a point on the curve we choose the coordinates yl.... , y'n and.represent the vector field W and the tangent vector i(t) in these coordinates:.

m m

W = ` yi. ^'y(t) = E^"`(t)yi

d

i=1 i=1

m A

E1'(t) ayi ('Y(t))i=1

Then

(07(r)W)(y(t)) = >dWA(7(t)) + [' ii(t) . W'(7(t)) . q(7m) 8

dt 8ykk=1

Along the curve y(t), this vector field is independent of the particular (lo-cal)extension of the vector field W(t). originally defined only on y(t), to

a vector field W. The formula thus uniquely defines a vector field VW/dt,lthe covariant derivative of W(t).

Definition 21. A vector field W(t) is called parallel along y(t) if

vW0.dt


In coordinates, this condition is equivalent to the following system of ordi-nary differential equations:

dWko.

14#

The existence and uniqueness theorem for ordinary differential equationsensures the existence of a unique parallel vector field W (t) for every initialvector W. E TAi a) at the initial point y(a) on the curve,

vwdt = 0. W(a) = Wa.

The value LV(b) at the end point ';f(b) of the curve is called the paralleldisplacement P., of the vector W. along the curve y(t),

P,(lV) := IV(b).Because of the linearity of the differential equation and the property

U(g(V,W)) = g(V V,W)+g(V,vuW)of the Levi-Civita connection, the parallel displacement

P.7:T.(,.)Alm Try(b)A.lm

is a linear isometric map between tangent spaces. The parallel displacementP.y,..Y2 along a curve formed by combining two curves y1 and y2, for whichthe end point of the first coincides with the initial point of the second, intoa single curve y1 * 72 leads to the same result as the superposition of thesingle parallel displacements,

P12 o P,., = P'Yl *'12

Consider a surface (1112, g) with a positive definite Riemannian metric anda closed curve y in M2 bounding a domain N2 C 1112. Parallel displacementalong y then is a rotation in the tangent space T.,(o)M2. The correspondingrotation angle can be computed from the Gaussian curvature.

Theorem 44. If the closed curve y bounds a domain N2 in a surface AI2with Gaussian curvature G, the parallel displacement along the curve is therotation through the angle

0 = JN2 r

Proof. Choose an orthonormal frame e1,e2 along -y in which e1 is tangentto the curve. In any parametrization y : [0, L] Ail of the curve, a vectorfield V = V1(s) e1(s) + V2(s) e2(s) is parallel if

dV1 dV2ds

= w12(e1) . V2 andds

= -w12(el) V1.


Consider the complex-valued function z : [0, L] C defined by z(s) :=Vl(s) + i. V2(s). Then we can write this system of ordinary differentialequations in the form

dzZ /= -2 W12(el) z.

The solution of this is

z(8) = z(0) exp l _ i J e Wl2(el) dsJ0

and, in particular, for the rotation angle of the parallel displacement weobtain

0 = - fr w12 = JN2 d'012 = JG.dM2.

Example 28. It is a peculiar property of the hyperbolic space 7.12 thatthe hyperbolic angle coincides with the usual euclidean angle. In general,the angle does not change under parallel displacement. Hence the paralleldisplacement of a vector along a geodesic line in a surface is the vector atthe end point of the line forming the same angle with the tangent vector asthe original vector did at the initial point. The points PQR in the picturebelow are the vertices of a geodesic triangle in the hyperbolic plane (seeExample 23). Start at P with any tangent vector (1), translate it to thepoint Q (2), then along the upper geodesic arc to R (3), and finally alongthe straight segment RP back to P (4). The resulting vector obviously doesnot coincide with the original one.

Parallel displacement allows one to generalize the covariant derivative V VTwith respect to vector fields to arbitrary tensors T. We explain this for ten-sors of type (k, 0). Assume that at a point a E Mm, the vectors W1.... W. E


TXM' are given, and let y(t) be the integral curve of the vector field V pass-ing at time t = 0 through the point x E Mm. Then we define

(VyT)(WI, ...,)4)(x) := dat(Ty(t)(Py(WI),

...,P7(Wk))It=o'

Here Py denotes parallel displacement along the curve y from the pointx = y(0) to the curve point -f(t). If, for example, T is a (1, 0)-tensor and Wa vector field, we obtain the formula

(VyT)(W)(x) = V(T(W))(x) - T(VyW)(x).

Theorem 45. The covariant derivative VyT of a (k, 0)-tensor is given by

(VvT)(Wi, ..., Wk) = V(T(Wi, ... , Wk)) - T(VvW1, W2, ... , Wk)- ... - T(W1i ... , Wk-1, VVWk) .

Remark. The metric g, viewed as a (2, 0)-tensor, is parallel:

(Vy9)(Wl, W2) = V(9(Wi, W2)) - 9(VvW1, W2) - g(WI, VvW22) = 0.

A completely elementary but somewhat lengthy computation, using theproperties of the Levi-Civita connection and the definition of the curvaturetensor, leads to an identity for the covariant derivative VTZ of the curvaturetensor, the so-called second Bianchi identity. We omit the simple proof andformulate only the result.

Theorem 46 (Second Bianchi Identity). Let U, V, W, W1, W2 be five vectorfields on the pseudo-Riemannian manifold (M', g). Then

(VuR)(V, W, W1, W2)+(Vy7)(W,U, W1, W2)+(VWR)(U, V, W1, W2) = 0.

Define the length of a curve y : [a, b] -+ Mm in a pseudo-Riemannian spaceby the integral

L(y) :=1 b I9( (t), ry(t)) I dt .a

For a variation yE : [a, b] - Mm of the original curve y(t) with fixed initialand end points, y. (a) = y(a),-yy(b) = y(b), the field of the variation,

W (t) (yE (t)) I E=o,

is a vector field defined along the curve y(t) vanishing at the end points. Itscovariant derivative VW/dt along the curve y(t) coincides with the covariantderivative Vyy(t)/de of the vector field ye(t) defined along the curve i7(c)-Y" (t),

dt (t) -V

de

t)

L-0


This formula is an immediate consequence of the local expressions for thetwo covariant derivatives. Parametrizing the curve y(t) by are length. weobtain the following formula for the variation of the length of the curve:

d(L(,E))IE_o =J

g(y t) ,y(t))dt1==0

= jbg(dw(t),ry(t))dt =lbg (W(t),

7- (t))dt.

Theorem 47. If the curve y : [a, b] Mm is a critical point of the lengthfunctional defined on the set of all curves with fixed initial and end points.then the tangent vector y: (t) is parallel along the curve y(t).

V7dt = 0.

Definition 22. A curve y(t) in the pseudo-Riemannian space (M"'.g)whose tangent, vector field is parallel is called a geodesic line.

Remark. If the curve y(t) = (y' (t), ..., ym(t)) is given in local coordinates,the equation for it to be a geodesic line is

d2yk M dy' dy'2 + Edt dt -;it

r = 0.*a=1

This is an ordinary differential equation of second order, and Corollary 3from §5.5 holds correspondingly:

Corollary 6. To each point x E M' and every vector V E TIMtm ona pseudo- Riemannian manifold without boundary there exists precisely onegeodesic line y,; : (-e, e) -+ Mm such that -y, (0) = x and i,(0) = V.

The geodesic line y (t) thereby determined is not necessarily defined forall values of the parameter t E R. The subset Ex C TxA1t of all vectorsV E TxMt for which exists for t = 1 is an open neighborhood of thezero vector in TTM". The exponential map at the point x of the pseudo-Riemannian manifold maps Ez to the base space,

Expx : Ex -p Mm, Expx(V) = yv(1)The exponential map is smooth, and its differential d(Expx) : To(T1M'") _T,Al"' -. TZMm is the identity,

dExpx(W) = dt(Exp(tW))It=o = dt(y'.w(1))It=" = W.

Hence the exponential map Expx is a diffeomorphism from an open neigh-borhood 0 E Ux C TZMm of the zero vector in the tangent space onto anopen neighborhood x E Vx C M^' of the point in the base space.


Simpler curvature invariants of the pseudo-Riemannian manifold arise fromthe curvature tensor by contraction (taking the trace), the most importantones being the Ricci tensor and the scalar curvature. The sectional curvatureappears as the Gaussian curvature of subplanes of the tangent space. l eshall now discuss these curvature quantities.

Definition 23. The Ricci tensor Ric is the first contraction of the curvaturetensor,

m

Ric(U, V) := F'g'"R(ei, U, V, e3) .

ij=1

The algebraic identities of the curvature tensor imply that Ric is a symmetric(2,0)-tensor,

Ric(U, V) = Ric(V,U).

Taking the trace of the Ricci tensor with respect to the pseudo-Riemannianmetric g (§5.4) leads to the scalar curvature r of the space.

Definition 24. The scalar curvature of a Riemannian manifold is the func-tion

T ._i j.k.l=1

ij9k1R(ei, ek, el, e3)

Contrary to the surface case (m = 2), the curvature tensor is no longer de-termined by the scalar curvature for manifolds of dimension at least three.Thus the Ricci tensor and the scalar curvature do not contain all the cur-vature information about a manifold. The last of the curvature quantitiesis the sectional curvature. Apart from the point of the manifold it depends,in addition, on a non-degenerate 2-plane in the tangent space. This is atwo-dimensional subspace E2 C ,,Mm whose orthogonal complement.

(E2)1 = {W E T Al- : g(V,W) = 0 for all V E E2},

is complementary to E2, E2 e (E2)1 = TXM. Intersecting E2 with anopen set 0 E U= C T=Ai"` on which the exponential map is defined and adiffeomorphism, we obtain a surface in M"',

M2(E2) := Expr(E2 n U:) .

Its tangent plane at the point x is E2, and, by restriction, the pseudo-Riemannian metric g of M'" induces a metric on M2(E2). The Gaussiancurvature G(x) of the latter at the point x E h12(E2) is called the sectionalcurvature K(x. E2) of the pseudo-Riemannian manifold at the point x inthe direction of the 2-plane E2. It can be computed using the curvaturetensor of Af '.


Theorem 48. Let V1, V2 be a basis of the non-degenerate plane E2. Then

1Z(V1, V2, V2, Vl)K(x, E2) --g(Vl, V1) g(V2, V2) - g2 (VI, V2)

Proof. For the proof, we assume that the metric of MI is positive definite.This is no essential restriction, but some of the formulas simplify. Note firstthat the right-band side of the formula for K(x, E2) does not depend onthe choice of the basis V1, V2; thus it suffices to prove this formula for anorthonormal basis in E2. Fix an orthonormal basis e1(x), e2(x), ... , e,,,(x)at the point x E C Al" such that the first two vectors belong to E2.We extend this basis to a local orthonormal frame by parallel displacementalong the geodesic straight segments {Expx(t W) : W E TXMm}. At thepoint x this frame is parallel,

Vei(x) = 0.Furthermore, on 1612(E2) the vector fields el, e2 are tangent to this surface,since M2(E2) consists of geodesic straight segments with initial vectors inE2. Let an, w,j be the forms dual to the frame e1, ..., e,,, and the connectionforms of the Riemannian manifold Mm, respectively. The embedding f :M2(E2) Mm allows us to restrict these forms to the surface,

A. := J'(aa), ti.i f'(wij)Then µ3 = p4 = ... = um = 0, and the structural equations of M'" imply,after restriction to M2(E2), that

t Ind{ll = ll2 t21, dA2 = n t12, X12 = Sln A Cat + n121 pI A 142-a

The pair µ1,/l2 is, within the surface M2(E2), the frame dual to el, e2, andthe last equations show that the connection form w12 of M2(E2) Coincideswith t;12, wit = e12. The Gaussian curvature a of M2(E2) is determinedby

- G - JAI A P2 = &w12 = d512 = tlnAta2+R2121 p1 AP2

At the point x E M2(E2), however, we have = g(De,e1, ea) = 0, andhence we obtain

- G(x) = R2121 = g(R(el, e2) el, e2) = - g(7Z(el, e2) e2, e1).

Special Riemannian manifolds are the so-called Einstein spaces and spacesof constant sectional curvature.

Definition 25. A pseudo-Riemannian manifold is called an Einstein spaceif the Ricci tensor is proportional to the metric, Ric = h g.


Remark. Contracting the equation Ric = h g leads to r = m h, i. e., thefactor of proportionality is uniquely determined by the scalar curvature ofthe manifold,

TRic = _19.m

We prove that this factor is constant.

Theorem 49. If (Mm. g) is a connected Einstein space of dimension m > 3.then the scalar curvature T is constant.

Proof. Choose an orthonormal frame el, ...,en and an arbitrary vectorfield U. After we insert the vector fields V = W2 := ei, W := ej, and W1ej into the second Bianchi identity and sum over all indices 1 < i, j < m,an elementary computation leads to the equation

> ej ((Vu)Ric(ej, ej) - 2(V Ric)(U, ei)) = 0.i

If the Ricci tensor is proportional to the metric, Ric = h g, the relationVg = 0 implies VuRic = U(h) g, and the last equation becomes

m U(h) - 2E ejej (h)g(U,ei) = 0.

Considering the vector fields U := ek, this yields (m - 2) dh(ek) = 0, and,in case m > 2, the differential dh of the function h vanishes. 0Example 29. In spherical coordinates, the standard metric of the two-dimensional sphere is the quadratic form

gJs2 = COS2(71') djl2 + d/12 .

Choose a pseudo-Riemannian metric h in the product R+ x R and a functionf : R+ x R R. On the four-dimensional manifold (1R3 -0) x lR = S2 x R+ x Rwe can then consider the rotationally symmetric metric

g := h (B e2f(r,t) . gJs2 .

A longer computation reveals that g is a four-dimensional Einstein metricwith vanishing scalar curvature if and only if the following pair of differentialequations is satisfied:

e-2f = A(f)+2lgrad(f)I2 and Odf.

Here, A and grad are the Laplacian and the gradient of the two-dimensionalpseudo- Riemannian manifold (R+ x Ift, h), G is its Gaussian curvature, and

Hess h(f)(V, W) := h(Vy(grad(f )), W)


is the symmetric Hessian form of the function f with respect to the metrich. Choosing, for example, the metric

r 1

h = Ll +2A1

Jdr2 + c dr C dt - c2 I 1 -

2Af 1dt2r r

111r J

and the function f (r, t) = log(r) gives rise to the Schwarzschild-Eddingtonmetric on (R3 - 0) x R, which, in Einstein's general theory of relativity,models the gravitational field generated by a mass concentrated at the pointx = 0. Further solutions of this system of differential equations describe thegravitational field of a rotating mass (Kerr metric).

Definition 26. A pseudo-Riemannian manifold (Mm,g) is called a spaceof constant sectional curvature (or space form) if there exists a functionK' : 111"' -+ R such that for every 2-plane E2 C TXMm

K(x, E2) = K'(x).

Theorem 50. The sectional curvature K = K' of every space Mmof dimension m > 3 is constant, and the curvature tensor is algebraicallydetermined by the metric,

R.(U,V)W = K'(g(V,W) -U -g(U,W) -V).

Proof. Consider the tensor

S(U, V, W1, W2)

= R(U, V, W1, W2) - K* (9(V, Wi)9(U, W2) - 9(U, W1)9(V, W2)),

and note that S has the same symmetry properties as the curvature tensor:

(1) S(U,V,W1,W2) = -S(V,U,W1,W2).

(2) S(U, V, W1, W2) = -S(V,U, W2, W1) .

(3) S(U,V,W1,W2)+S(V,W1,U,W2)+S(W1,U,V,W2) = 0.

(4) S(U, V, W1i W2) = S(W1, W2,U, V) .

An additional property of the tensor S results from the formula for thesectional curvature:

(5) S(U, V, V, W) = 0 .

A calculation only making use of equations (1) - (5) shows that this tensorhas to vanish, S - 0. In fact, from

0 = S(U. V + W2, V + W2iU)= S(U. V. V, U) + S(U, V, W2, U) + S(U, W2, V, U) + S(U. W2, W2, U)

= 2 - S(U, V, W2, U)


we first conclude that S(U, V, W2, U) = 0. Hence we have succeeded inreplacing the third argument. V in the identity (5) by an arbitrary vectorW2. We repeat this step once more, starting from S(U, V, W2, U) = 0. Thus,as the fourth entry another independent vector W1 can be substituted forthe vector U, that is.. S vanishes, and the curvature tensor of a space formis determined by

R(U, V)W = K`(g(V, W) U - g(U, W) V).Contracting the curvature tensor, the resulting Ricci tensor is proportionalto the metric,

Ric = (m-1) K*.g.Hence M"' is an Einstein space, and K* is a constant function. 0

The components Ra31 of the curvature tensor of a space form with positivedefinite metric (for simplicity) are

Ra3i = K' - (b3idaj - 63j6ai),and the structural equations simplify-,

m

do'i = Qa A Wai dwij = E w1a A ula j - K* ai A o'j.a=1 a=1

In the middle of the 19th century. Riemann gave the complete local descrip-tion of these spaces.

Theorem 51 (Riemann, 1854). Each space form is locally isometric to thespherical space S'", the euclidian space R"', or the hyperbolic space 1.tm.


Exercises

1. Let y(s) be a curve in its natural parametrization, and let p(t) be thissame curve in an arbitrary parametrization, which hence is related to 7 viaEa(t) = (s(t))-

a) Prove the following formula for the second derivative of p:

d2µ(t) _ d27 ds 2 dy d2s = 2µ(t) = dt2 182 (dt + ds dt2(t) (s) h + s t .

b) Deduce from this the formula for the curvature in a general parametri-zation:

mo(t) = IIi(t) X µ(011IIi (t)113

c) Prove that the torsion can be written in the Frenet frame as

r(t) (Fx h', h) = det (t, h, h') = det (7', 7", -y"')K(t)2

d) Derive in an analogous way to a) a formula for -ii (t), and conclude thatthe torsion is given in an arbitrary parametrization by

r(t) =det (µ(t), µ(t), i (t))

IIi (t) x A(t)112

2. Let C C R3 be a curve all of whose tangents pass through one and thesame point. Prove that C is part of a straight line.

3. Let C C 1123 be a curve all of whose tangents are parallel to one and thesame straight line. Prove that C is part of a straight line.

4. Show that the tractrix (Example 11) is the curve passing through thepoint (1.0) on the horizontal axis with the property that the length of thesegment of the tangent line from any point on the curve to the vertical axisis constant. The following intuitive interpretation gave it its German name"Schleppkurve"(tow curve): It is the path that a dog pulling on a leash tothe west is constrained to take when his master walks along a north-southpath.

5. Compute the curvature and the torsion of the following curves:

a) y(t) = (t. a(e'/° + e-t/a)/2, 0) (catenary);

Exercises 199

b) -y(t) = (a cost, a sin t, b t) (helix);

c) y(t) = (a(t - sint), a(l - cost), bt);

d) y(t) = (t, t2, t3).

6. Prove that y(t) = (at, b t2, t3) with a, b > 0 is a slope line if and only if2b2 = 3a.

7. Prove that a C3-curve C is a helix if and only if its curvature r, > 0 andits torsion r are constant.

8. Determine all plane curves C C R2 satisfying

a) K(s) = const;

b) c(s) = 1

c) K(s) =

9. The Darboux vector of a curve with non-vanishing curvature is the vectorC T:= r Prove that the Frenet formulas can be written in the followingform:

ds = d x t,ds = j -X hh,

ds= d *X 6.

10. Consider the spherical curve y*(s) := t(s) : [0, L] S2 C iR3 andsuppose that the curvature K(s) of the original curve y(s) does not vanish.Prove that 'y' (s) is a regular curve, and compute its curvature K' and itstorsion r'. Which curve y*(s) arises in the case of the helix y(s)?

11. A slope line with slope angle a lies on a sphere of radius R if and onlyif its curvature K(s) and its arc length s satisfy the relation

S2

2

tan2 a + K2 (S)_ R

12. Let y(s) : [0, L] -> R2 = C be a closed plane curve, and let p be a pointin the plane not belonging to the curve. Prove that the integral

1 jL t(s)ds

21ri -Y(s) - Pis an integer. This is called the index or winding number of the curve y(s)around the point p.


13 (Jacobi, 1842). Let -y C R3 be a closed curve with non-vanishing curva-ture, and suppose that the curve of principal normal vectors h is a simpleclosed curve in the sphere S2. Prove that h divides the sphere into two partsof equal area.

14. Classify all surfaces of revolution with constant Gaussian curvature us-ing the results from Examples 3 and 11. Altogether there are 9 geometrictypes. Which of these can be compact manifolds without boundary?

15. Let F : U -* 1R3 be a parametrized surface such that the tangent vectorsr9/au and a/av are orthogonal at each point (u,v) E U. Set

E _ \ 49u, au. > , C = \8v49 a

' Ovand choose the orthonormal frame

1 8 1 ae

_ au' e2 _ av

a) In the dual frame al = /Edu, 02 = /dv, the connection form w12 isgiven by

wl2 =1 aV G al +vE - av &

b) The Gaussian curvature is equal to

a a (8,/-G/8u)jK = - EG av +OU Ec) As an application, determine the Gaussian curvature of a surface of

revolution whose generating curve is not necessarily parametrized byarc length.

16. Let 1112 C 11 P3 be a surface with non-vanishing Gaussian curvature.Prove:

a) There exist two orthogonal vector fields V and W with lengths IIVII =1 and IIWII = IHI whose flows are volume-preserving, Gy(dM2) =Gw(dM2) = 0.

b) The integral curves of the vector field W/IH) are geodesic lines.

c) Let B : M2 - R be the function measuring the geodesic curvature ofthe integral curve of the vector field V at a point. Then W(H) = H2 B.

d) The following commutator formula holds: [W/IHI, V) = B V.

Exercises 201

17. Discuss the geometric properties of the helicoid and the catenoid, andstudy the map transforming the one into the other.

18. Enneper's minimal surface is parametrized by (z = a + ib)

F(a, b) = (Re(3z - z3), Re i(3z + z3), Re(-3z2)) .

Compute its Gaussian curvature. Prove, moreover, that its curvature lines(i. e., the curves on the surface obtained by fixing one of the two parametersa or b) all lie in a plane through the y- and z-axis and have polynomial arclength. Determine the equation for the tangent plane to a point x E F andtransform it into

(p - q, x) = (p, p) - (q, q)This is the equation of a plane with respect to which the two points p andq are mirror images of one another. The curves p(a), q(b) are parabolas inthe perpendicular planes x = 0, y = 0, and the summit of each of themcoincides with the focus of the other (so-called focal parabolas). Illustratethese properties using Maple or Mathematica.

19. The minimal surface studied by Bour (1862) arises from the Weierstrassrepresentation by inserting f (z) = czm, g(z) = z. Prove that it can bemapped to a surface of revolution in a length-preserving way.

20. Prove the following properties of the modular surface of a holomorphicfunction f : U -* C:

a) The modular surface is hyperbolically curved in a sufficiently smallneighborhood of a finite pole of f.

b) Let zo be a finite zero of f . If this zero is multiple, the modular surface iselliptically curved in a sufficiently small neighborhood of zo. If the zerois simple, every sufficiently small neighborhood of zo decomposes into2k - 2 sector-like pieces, in which the modular surface is alternatinglycurved elliptically and hyperbolically. Here k is the exponent of thenon-vanishing term succeeding the summand a, (z - zo) in the seriesexpansion of f.

In the next two exercises it will be helpful to use the position functions pifrom the proof of the Minkowski-Steiner theorem.

21. Let M2 be a surface with normal vector e3. Prove that the 1-form77 1 = (x, de3) defined on the surface is closed, and conclude that, for acurve ry C M2, the relation

j(xde3) = 0


holds if one of the following two conditions is satisfied:

a) The curve y is the boundary of a two-dimensional submanifold G2 C M2.

b) 111 is simply connected.

22. Let A12 be a compact surface without boundary. Prove that there existsa point ro E A!2 with the following properties:

a) The normal vector e3 is parallel to the position vector rO at the pointro.

b) The Gaussian curvature is positive at xo, G(xo) > 0.

Hint: The function f : M2 -+ R, f (x) = (x, x), has to have a critical pointon Ate.

23. Let the group G = SL(2,R) act on the hyperbolic plane ?{2 by

a b az+b6 F

_c d

.z _cz+d'

Verify that the image point g z actually does belong to ?{2, and that super-position of two of these transformations corresponds to matrix multiplicationin G. Show. moreover, that each g E G leaves the metric invariant; hence Gis an 3-dimensional group of isometries from N2 onto itself.

24. Let AI'; = 1R3 and denote by DyX the directional derivative of thevector-valued function X : R3 R3 in the direction of the vector X. Provethat

VXY := DXY+2X xYdefines a covariant derivative having all the properties (1)-(4) from Theorem42. but violating property (5).

25. Consider on the set {(x,y) E R2 : -7r/2 < y < 7r/2} the pseudo-Riemannian metric

1 _ dx2-dyeCOS2

x'2 - y'2 x'a) Prove that E _2(y) and P =2(y) are first integrals of the

cosgeodesic flow, satisfying, in addition, the inequality p2 - E > 0.

b) Discuss the geodesic lines on M2. To do so, assume that y is a functionof r and integrate the resulting ordinary differential equation.

c) On Al" there exist points that cannot be joined by a geodesic line.

Exercises 203

26. Prove that every three-dimensional Einstein space is a space of constantcurvature.

27. Let M' be a non-flat Einstein space (for example, S"'). Show thatMI x M"' with the product metric is an Einstein space, but not a space ofconstant curvature.

28. It is well-known that a symmetric bilinear form h.(x, y) on a vectorspace is completely determined by its quadratic form q(x) := h(x, x) (via thepolarization formula: 2h(x, y) = q(x + y) - q(x) - q(y)). Prove, in a similarway, that the Riemannian curvature tensor 1 (U. V, W1, W2) is completelydetermined by the quadratic form K(U, V) := R(U, V, V, U) correspondingto the sectional curvature.

29. Prove that a four-dimensional Riemannian manifold is an Einstein spaceif and only if for every 2-plane E2 C TM4 and its orthogonal complement(E2)1 the corresponding sectional curvatures coincide, K(E2) = K((E2)1).

30. Because of its symmetry properties, the curvature tensor of a pseudo-Riemannian manifold can be interpreted as a transformation R : A2(M'') -A2 (M7°) on 2-forms,

1n

R(ai Aaj) = 2 > Ro3i . a, Aa,3.Q,:3=1

In dimension m = 4, this gives rise to an endomorphism R : A2(A14)A2(M4). On the other hand, the Hodge operator * : A2(A14) A2(M4)acts on A2(M4), and its square depends only on the index k of the metric (seeTheorem 5, Chapter 1): ** = (-1)k. In the cases k = 0 (positive definitemetric) and k = 2 (neutral metric), the Hodge operator decomposes the realbundle A2(M4) into the corresponding eigenspaces A2 (M4) (see Exercise8, Chapter 1). Prove that the block representation of the curvature tensor.

_ R++ R-+R R+- R--with respect to this decomposition of A2(M4) has the following properties:

a) 7Z is symmetric, i.e., R++ = R++, R__ = R__ and R+_ = R_+.

b) The traces of R++ and R__ coincide, tr(R++) = tr(R_-) = -r/12.

c) M4 is an Einstein space if and only if R+_ vanishes.


Literature: Th. Friedrich, Self-duality of Riemannian manifolds and connec-tions, in: Riemannian geometry and instantons, Teubner-Verlag, Leipzig,1981, 56-104.

The Einstein equation in the general theory of relativity combines the geo-metric curvature quantities of a four-dimensional pseudo-Riemannian man-ifold of signature (1, 3) with its physical properties encoded in the energy-momentum tensor T,

Ric- rcT.

Here Ric. g. r are the Ricci tensor, the metric, and the scalar curvature;K is a constant depending on the chosen system of units. Already for thevacuum, T = 0, there are non-trivial, physically very interesting solutions ofthis equation, among others the Schwarzschild metric to be discussed now.Obviously, a vacuum solution of the Einstein equation has to be an Einsteinspace with vanishing Ricci tensor in the sense of the definition given before.

31 (Schwarzschild metric). On a spherically symmetric and static space-timemanifold M4 (this is a pseudo-Riemannian manifold of signature (1, 3) withisometry group SO(3, R) and a distinguished time direction), it is alwayspossible to introduce coordinates from R x R+ x S2 with respect to whichthe metric can be written as

[e 2b(r) dr2 + r2 (d92 + sin2 O dV2)J .g = e2a(r) dt2 -

Here, the functions a(r) and b(r) asymptotically tend to zero for r - co(the metric is "asymptotically flat"). We introduce the following basis of1-forms:

ao = ea dt, Cl = eb dr, a2 = rdO, 03 = r sin 9 dcp .

a) Check that the metric satisfies g = 0,02 - 0i - 02 - a3 .b) Compute the forms dai and show, using the first structural equation,

that the connection forms are given by

wo, = -a'e-bcO, WO2 = W03 = 0,

e-b a-b cot 9W12 = - 0`2, w13 = - a3, w23 = a3-r r r

c) We introduce the notation lid := 2 E.4 Rtasia0 A0,3. (f is the so-calledcurvature form). Show, using the second structural equation, that

110 = e-2b(a'b'-a"-a2)aoAal, f20 = -a'er"aoAa2,

03O =-a'er6

ao A 03, 1121 = bier6a1

A02,

5231 = b'erbal A a3, X32 = '-'-'012 A a3,

Exercises 205

and compute from these formulas the components Rkl of the curvaturetensor in this basis.

d) Compute the components of the Einstein tensor G := Ric - 1g ,r:

Goo= I - e-2b (13 _ 2b ) , Gi l = ,1-s _e-2b(77 + ) ,7- F

G22 = G33 = -e-2b (a'2 - a'N + a-+ =d)r

e) Solve the vacuum equation by means of this Ansatz. The result is (Mis a constant of integration)

r 1 r 1_1

g = 11- 2MJ

dt2 - 11- 2M

Jdr2 - r2 (d92 + sin2 0 d 2)

r I r32. Restrict the Schwarzschild metric g to the two-dimensional submanifolddefined by 0 = x/2 and t = const. Prove that this yields the metric ofa paraboloid of revolution (a "lying" parabola!) with the equation z2 =8M(r - 2M) (see the picture).

33. Light moves along geodesic lines whose tangent vectors have length zero.Making use of the fact that the equations defining a geodesic are preciselythe Euler-Lagrange equations of the length function G = ta gjjx'ia, provethe following assertions:

a) A particle moving at t = 0 in the equatorial plane 0 = x/2 stays thereforever.

b) The quantities L := r2cp and E := t(1-2M/r) are first integrals. More-over, the second Kepler law holds: The orbit ray covers equal areas inequal times.


c) Set r = r(W) and derive the equation describing the motion of light rays.Result: It is reasonable to set u := 1/r: u" + u = 3Mu2.

d) Solve this differential equation approximately up to second order in V.Result:

uo = 1sin V +

2 0

[1 +

3

cos 2W .

Interpret the constant of integration ro as the scattering length. Whichasymptotic value arises for cp if r tends to oo and sp is supposed to staysmall? Twice this value, denoted by 5 in the picture, is the relativisticdeviation of light in the gravitational field of a very large miss, whichcan be described by the Schwarzschild metric.

Chapter 6

Lie Groups andHomogeneous Spaces

6.1. Lie Groups and Lie Algebras

In the preceding chapter Noether's theorem showed that symmetry consider-ations simplify the study of geometric problems, and sometimes it is only bysymmetry considerations that a solution is possible at all. In fact, beginningin the 1870s, the conviction grew that the basic principle organizing geom-etry ought to be the study of its symmetry groups. In his inaugural lectureat the University of Erlangen, which later became known as the "ErlangerProgramm", Felix Klein said, in 1872,

"Es ist eine Mannigfaltigkeit and in derselben eine Trans-formationsgruppe gegeben; man soil die der Mannigfaltigkeitangehorigen Gebilde hinsichtlich solcher Eigenschaften un-tersuchen, die durch die Transformationen der Gruppenicht geandert werdenl.

One has to distinguish whether the groups under consideration are dis-crete (for example permutation groups) or continuous (for example, one-parameter groups of isometries). The latter were systematically introducedby the Norwegian mathematician Sophus Lie (1842-1899). which is why theybear his name today.

1 "Let a manifold and a transformation group in it be given; the objects belonging to themanifold ought to be studied with respect to those properties which are not changed by the trans-formations of the group."-quoted from F. Klein, Des Erlanger Programm, Ostwalds Klassikerder exakten Wissenschaften, Band 253. Verlag H. Deutsch, Frankfurt a. M., 1995, p. 34.

207

208 6. Lie Groups and Homogeneous Spaces

The fundamental idea of a Lie group is a very simple one. It ought to be agroup which is at the same time a manifold, and hence allows a differentialcalculus. Moreover, the manifold structure has to be compatible with thegroup structure, i. e., the product is a differentiable map.

Definition 1. A Lie group is a group G which, at the same time, is adifferentiable manifold such that the map

is (infinitely often) differentiable.

Remark. Obviously, the last condition is equivalent to requiring that theproduct map and the inversion

(9, h) -. g . h, g'-' g-1 ,be differentiable maps.

Example 1. Every finite-dimensional vector space V is an abelian Lie groupwhose group product is exactly the addition of vectors (v, w) H v + w.

Example 2. The unit circle S' = {z E C : IzI = 1} is an abelian Lie groupwith the usual multiplication of complex numbers as product.

Example 3. Most Lie groups can be realized as matrix groups. The set ofall invertible matrices with entries in K = R or C is an open subset of IK"zand hence a manifold. Endowed with matrix multiplication as product, itforms a Lie group, the general linear group

GL(n,K) := {A E .M"(K) : det A 540} .

More generally, GL(V) is meant to denote the group of invertible endomor-phisms of the vector space V.

Example 4. The vector space K" and the general linear group GL(n, K)combine to form a new Lie group, the affine group Aff (K") := GL(n, K) x IIS"with multiplication

(A, v) (B, w) := (AB, Aw + v).This product rule arises in a natural way by defining an action of the affinegroup on the vector space K" through

Aff(K") x K' V. (A, v)x := Ax + v.and then applying the transformations determined by (B, w) and (A, v) ona vector x one after the other.

Each Lie group G acts on itself by means of the left and the right translationwith a fixed element g E G,

L9, R9 : G -. G, L9(h) = g h and R9(h) = h g.

6.1. Lie Groups and Lie Algebras 209

The corresponding differentials are the following maps in the tangent bundleof G:

(dLg)h: ThG - TghG, (dR9)h : ThG Th9G.

To avoid double indices, in this chapter we will use the notation (df)h insteadof f.,h for the differential of a map f.

Definition 2. A vector field X on G is called left-invariant (or right-invariant, respectively) if it is transformed into itself by dL9 (or dR9, i.e.,dL9X = X. At a point h E G, this means

(dLg)hX(h) = X(g - h).

Since left translation is obviously a diffeomorphism of G, Theorem 35 in§3.9 can be applied to yield, for the commutator of two left-invariant vectorfields, the formula

dLg[X,Y] = [dL9X,dLgy] = [X,Y1.

This property, together with the fact that vector fields satisfy the Jacobiidentity (Theorem 34, §3.9), endows the vector space of left-invariant vectorfields with the structure of a Lie algebra, the Lie algebra g of the Lie groupG.

Theorem 1. The vector space of left-invariant vector fields on a Lie groupG is canonically isomorphic to the tangent space at the neutral element,

g='TeG.

Proof. With each left-invariant vector field X, we associate its evaluationat the neutral element e, X - X(e) =: X E TOG. Conversely, every elementX E TeG determines a vector field Xx on G by setting

Mx(g) := (dL9)e(X)This satisfies the relation

Xx(gh) = (dLgh)e(X) = (dLg)h(dLh)e(X) = (dLg)hXX(h)hence Xx is left-invariant. 0Remark. Because of this fact, we will no longer distinguish between theLie algebra of left-invariant vector fields and the tangent space to the groupat the neutral element. Its elements will be denoted by upper case Latinletters X, Y, ... E 9.

Choosing a basis X1, ..., Xr of 9, we can again write their commutators aslinear combinations of the basis elements,

r

[Xi,Xj] = &Xk-k=1


The antisymmetry of the commutator and the Jacobi identity imply thatthe constants C have to satisfy the relations

C - -C ' CijCk,n + Cj' "ki + CrniCkj = 0.ij iiFollowing E. Cartan, the numbers C are called the structure constants ofthe Lie group G, since Cartan's structural equations are simple to formulatein their terms. To see this, we agree to call a differential form w on Gleft-invariant if it satisfies the condition

L*9w = w.

Following the argument in the case of vector fields, it is easy to see thatthe r-dimensional vector space g* of left-invariant 1-forms is canonicallyisomorphic to Te G. Now let al, ...,o-, be the basis of g* dual to X1. ..., Xr.

Theorem 2 (Maurer-Cartan Equations). Let C be the structure constantsof a Lie group G with respect to the basis X1, ... , Xr of its Lie algebra g.Then the exterior derivatives of the forms in the dual basis al, ... , or of g*are given by

do, _ -r k of Aak.j<k

Proof. From Theorem 33, §3.9, we know the general formula for the differ-ential of a 1-form oi:

do (Xj, Xk) = Xj (0t(Xk)) - Xk(ai(Xj)) - ai([Xj, Xk))Since the forms ai are dual to the basis vectors Xj, the first two terms onthe right-hand side vanish. Now

dn'i(Xj,Xk) = -ai([Xj,Xk]) = -ai(E,nC7kXm) = -Cjik,which is precisely the assertion.

We introduce a 1-form e : T(G) -+ g defined by the formula

e(t) := dL9-1(t)for every tangent vector t E T9(G) at the element g E G. The form eshifts the tangent vectors of the Lie group via left translations to the spaceTe(G) = g. It is the so-called Maurer-Cartan form of the Lie group. Ingeneral, if el, e2 are two 1-forms with values in the Lie algebra g, we defineits "exterior" product as a 2-form with values in the Lie algebra by theformula

[ei,e21(tl,t2) [el(tl),e2(t2)] - [el(t2),e2(tl)lIn particular,

[e, el(tl, t2) = 2 - [9(tl ), e(t2)],


and the Maurer-Cartan equations can be formulated equivalently as

0.

Example 5. The special linear group in dimension two,

SL(2, R) = {A E M2 (R) : det A = 1 } ,

is a 3-dimensional Lie group with Lie algebra (see Exercise 5, Chapter 3)

s1(2, R) = {X E M2(R) : tr A = 0 } .

The matrices

H =10

011' E [0 01' F [1 0,form a basis of sl(2, lR) and satisfy the comniutator relations

[H, E] = 2E. [H, F] = -2F, [E, F] = H.If we denote the forms of the dual basis of sl(2, I8)` by ax, 0E, and aF, theMaurer-Cartan structural equations read as

dvy = - CE A QF, dOE = -2 aH A O'E, doF = 2 otl A aF .

Example 6. The Lie group GL(n, K) is an open subset of M. (K). Its Liealgebra is therefore

gl(n,K) '=' Te GL(n, K) = M,. (K) .

Theorem 3. Let X be a left-invariant vector field on G and denote by 4x (t)the maximal integral curve of X such that 4x (0) = e. Then

(1) 4x(t) is a group homomorphism. 4x(t1+t2) = 4'x(tt) ° 4>x(t2)

(2) the vector field X is complete, i. e.. 4b,y(1) is defined on all of R.

Proof. Let 4bx(t) be defined on the interval to < t < tb. We start byshowing the first assertion for two values tl, t2 such that they together withtheir sum t t + t2 belong to the domain of definition of 4x (t). To do so, fixt 1, set. g 4,x (t 1), and consider the curve

z1(s) = g 4x(s) on the interval (ta, tb)as well as the integral curve obtained by shifting the argument by tt,

zit (s) = Ix(tt + s) on the interval (t,i - tt, tb - tt) .

From their derivatives,dzi ,r fd4x 1 r /-'- , \ v/ . , ". v/ , %%

d

anddill __ d4bx(ti +s)ds ds

= X (tx(tt + s)) = X (zlt (s)),


we conclude that q and ill both are integral curves of X with equal initialconditions.

fi(0) = 'x(ti) = g = 9(0)Hence'i and r11 have to coincide on (ta, tb) fl (ta - t1, tb - t1). One easilychecks that t2 has to belong to this intersection, i. e..

' x(tl+t2) = 0 o4x(t2)Now we show that tb = 00; to prove ta = -oo one proceeds analogously.Suppose that tb is finite. Then,

r)(s) _ (px (2) x (s -2

)

is a curve defined on [ta + tb/2, 3 tb/2)) with tangent vector

ds = 2) = X (sx() 4x(s - 2 = X(gl(s)).

Hence ()(.s) is an integral curve of X defined on [ta + tb/2, 3 tb/2) such thatq(s) = 4 x (s) for s E (ta, tb) fl [ta + tb/2, 3 tb/2). Thus the maximal integralcurve has to be defined at least on 3 tb/2, a contradiction to the maximalityof tb.

This theorem enables us to define the exponential map for Lie groups.

Definition 3. The exponential map exp : g -. G is the evaluation at t = 1of the integral curve 4 x of X E g satisfying 4x (0) = e,

exp(X) 4x(l)Theorem 4. The exponential exp : g -y G is a smooth map with the fol-lowing properties:

(1) exp(O) = e;

(2) exp(-X) = [exp(X)]-1;

(3) exp ((s + t)X) = exp(sX) exp(tX).

In addition, it is a diffeomorphism from an open neighborhood V of zero ing onto an open neighborhood of the neutral element in G.

Proof. For the smoothness claim, we have to refer to the fact that thesolution of a system of differential equations depends smoothly on the initialconditions. Under scalar multiplication, the flow transforms as

Op (t) = 4Dx(A-t),

sinced4x(At)

= a X('Px(,\t))dt


together with'x(A 0) = e implies that cx(A t) is an integral curve ofA X. Restricting the exponential map to a straight line through the originyields a one-parameter subgroup of G. This proves property (3):

exp ((s+t)X) = 1D(s+t)x(1) = 4x(s+t) =4tx(1) = exp(tX) exp(tX).

Property (2) is an immediate consequence of (3). and property (1) is trivial.The final assertion follows from the inverse function theorem if we can showthat the differential of the exponential map at the origin,

d(exp)o: Tog=9- TG=9.is invertible. But this is just the identity map: Consider the curve -Y(s) _s X in g with tangent vector 7(0) = X. Then

dexp(sX) = d48x(1) _ dqbx(s)d(exp)o(s.X) = ds 18=o ds I3=0 ds I 8=0 = X(e).

Since we already know that the evaluation of a left-invariant vector field atthe neutral element e is an isomorphism, the proof is completed.

It turns out that all one-parameter subgroups of a Lie group G have theform exp(tX) for some X E 9.

Theorem 5. Every continuous group homomorphism ' : R G has theform

11(t) = exp(tX)

for a certain X E 9. In particular, every continuous group homomorphism1R G is smooth.

Proof. Let 41 : 1R G be a continuous group homomorphism, and letV C g be a neighborhood of the origin so small that exp : g - G is adiffeomorphism on 2V. Let 11' = exp(V) be the image of V in G. Since %Fis continuous, there exists an e > 0 such that 'P([-e, e]) C W. In particular,fle) has to have a pre-image X E V under the exponential map, %P(e) _exp(X). The smooth group homomorphism

(ix).f : G, f(t) = exp

coincides with 'F(e) at the point . Consider the set

K = {tEIR: 'F(t)= f(t)}.

This is a closed subgroup of IR, since ' and f are continuous group homo-morphisms. Therefore, K is equal either to R or to a discrete subgroup of it.


The latter case will be excluded, implying the equality of f and i. Assumefor a contradiction that K is generated by a number a > 0,

K = In-a: nEZ}.As c already belongs to K, a has to be less than e, and '+(a) belongs to W.«'e show a/2 E K to obtain the desired contradiction. To do so. start fromthe Ansatz

f (a/2)2 = f(a) = 'I'(a) = 'I' (a/2)2 .Then, applying the inverse of the exponential map,

2exp-' (f (a/2)) = 2exp 1('' (a/2))

and noting that'P(a/2) belongs to IV, we conclude that 2exp-' (1P (a/2)) E2V. On 2V. the exponential map is a diffeomorphism, and hence

f (a/2) = IP (a/2) .

This means that a/2 E K. 0Definition 4. Let G1, C2 be two Lie groups, and let 0 : G1 -. C2 be agroup homomorphism. We define the differential of io as a map between theLie algebras of left-invariant vector fields on GI and G2 as follows: Considerfor X E g i the one-parameter subgroup exp(tX) E G1. By assumption,its image curve '(exp(tX)) is a one-parameter subgroup of G2, which byTheorem 5 has to have the form exp(tY) for some Y E 92. Now we set

u' : 91 - 92, y.(X) := Y.

This definition immediately implies that the following diagram commutes:

92

jex

C2

Identifying 91 - TeG1 as well as 92 ^_' TeG2, we see that this definitionis compatible with the usual one of the differential at the neutral element,di'e : TeG1 -+ TE.G2. More precisely, the diagram extended by d,', remainscommutative:

91v.

exp

G1

6.2. Closed Subgroups and Homogeneous Spaces 215

To see this, take any left-invariant vector field X E gl and compute_ dexp(tO.(X)) _ dV,(exp(tX)) _.(X)(e) dt I _ dt dVe(X (e))

=o t=oA further important property of the differential V. is its compatibility withthe commutator.

Theorem 6. The differential of a homomorphism between Lie groups, :

GI -+ G2. is a homomorphism of Lie algebras, i. e..

14'.[X.Y] = [?P.X,V'.YJ-

Proof. The commutativity of the diagram just discussed is equivalent tothe following identity, holding for every left-invariant vector field X E g:

iP.(X)(e) = dl/'e(X(e)).From this, we deduce the analogous identity holding at each point g E GI,

d,;"9(X (g)) = dt''g(dLg)e(X (e)) = d(it' o L9)e(X(e)) = (dLg9))ed ke(X (e))

_ (dL, ls))e (X) (e) = w'.(X)(vI(g)).

This means that the vector fields X and ,b.X are ?P-related in the sense ofthe remark following Theorem 35, §3.9. In particular, this remark impliesthe assertion.

6.2. Closed Subgroups and Homogeneous Spaces

The exponential map of Lie groups does not satisfy the functional equationfamiliar from calculus; it turns out that a second order correction has to bemade.

Lemma 1. Let G be a Lie group and let exp : g G be the exponential map.Then, for two arbitrary left-invariant vector fields X, Y E g, the followingidentity holds:

exp(tX) exp(tY) = exp (t(X + Y) + 0(t2))

Here 0(t2) indicates that 0(t2)/t2 stays bounded for t - 0.

Proof. Choose a neighborhood V C g, 0 E V. such that exp : V - W' CG is a chart around the neutral element e E G, and also choose a basisXI, ... , Xr of g. By means of

(XI , ... , xr) _ exp(x1Xi +... + xrXr) E G,we introduce coordinates in W. Let U C W be so small that U U C W.(By " " we denote the product in the group G.) On U x U C G x G. wethen have the coordinates

y' = x'opri, z' = xtopre


and, because U U C W, we can restrict the group multiplication to obtaina mapp:UxU-+W. Let

i i 1 r 1 r

be the coordinate representation of p. Then

f`(0, ...,0,z1, ...,zr) = z', f'(yl, ...,yr,0, ...,0) = y',and hence f' has the Taylor expansion

i i

ft(y..) = f`(0,0)+ k(0,0)yk+ ayl(0,0)zl+R,

where the remainder R consists of second order terms. Inserting tY =y'Xi. tZ = z'Xi into the Taylor expansion, we now obtain

exp(tY + tZ + R) = exp(tY) exp(tZ),

and the term R tends to zero like O(t2). O

The following example shows that not every continuous subgroup of a Liegroup has to be a Lie group itself.

Example 7. Let T2 = Sl x S' be the two-dimensional torus, parametrizedby (t. s) E [0, 1] x [0, 1]. For every real number q, the formula yq : R -T2, yq(t) = (t, q t) defines a curve in T2, which is isomorphic to the additivegroup R of real numbers. We want to show that this curve is dense in T2 forirrational q. and hence provides an example of a connected, non-closed one-parameter subgroup of a Lie group. Let (to, so) E [0,1]2 be an arbitrarilygiven point on T2, which we want to approximate by a sequence of pointsin yq. For each integer n E Z, we have

,y(to+n) = (to+n,qto+qn),and, as an angle coordinate in S1, to + n is equivalent to to. Similarly,qto + qn + m is equivalent to qto+qn for every integer m; hence it correspondsto the same point on yq and on T2. If we are able to show now that qZ + Zis dense in R. then we could conclude the existence of a sequence of pairs(n;, m,) E Z x Z for which

limgni+mi = so-qto.i-00

But then q(to + ni) + mi converges to so for i - oc, and

lim ryq(to + nt) = 'Y(to, so),

as claimed. Thus it remains to prove the following lemma.

Lemma 2. For every irrational number q, the set Z + qZ is dense in R.


Proof. Like the set Z+qZ, its closure is a subgroup of R. The group Z+qZcannot be cyclic; if it were, it would be generated by an element a > 0, andthere would exist integers k and I such that

I = q = Va.But this would imply q = Ilk E Q. Hence we arrive at a contradiction. Asthe closed subgroups of IR are precisely the cyclic groups and R itself, onlythis last possibility remains for Z + qZ.

In Example 4, §3.1. we encountered a parametrization of a torus of revolutionby means of S' x S'. The following pictures show the trace of a curve yyin this parametrization for two close rational and irrational values of q,respectively. In Example 8, §7.4, we show that the motion of a sphericalpendulum can be parametrized precisely by such a curve on a torus.

The main objective of this section is to prove that a closed subgroup H of aLie group G is a submanifold of it. Consequently, it is itself a Lie group, andthe quotient space C/H carries the structure of a manifold with a smoothG-action. As a preparation, we need a few technical lemmas. Let II - II beany norm on the vector space g.

Lemma 3. Let H be a closed subgroup of G. and let Xn # 0 form a sequenceconverging to zero in 9 such that exp(X,,) belongs to H and Xn/IIXnII tendsto an element X E g. Then

exp(tX) E H for all t E R.

Proof. For a fixed t > 0 we define a sequence of natural numbers m by

mn := max{kEN:Then the following estimate holds:

mnIIXnII < t < (mn + 1)IIXnII = mnIIXnII + IIXnIIBut on the right-hand side, the sequence IIXnII tends to zero, and hence

lim mnIIXnII = t.n-or,


This implies

limn rnXn = hn-oo m m HIX,,IIIlXnll = t X ,

and, since the exponential map is continuous,

Iim exp(t X).n-ooEach term in the sequence exp(m,,Xn) = [exp(X,,)]"'" is an element of H.As H is closed by assumption, the limit of this sequence, exp(t X), has tolie in H, too. The proof for the case t < 0 proceeds along the same lines.

Lemma 4. For every closed subgroup H C G, the set

b :_ {XEg: exp(tX)EHforalltElR}is a linear subspace of g.

Proof. For any vector X E h, its scalar multiples a X also belong to h. Itthus suffices to show that h is closed under addition. To this end, let X, Ybe elements of h, and suppose that X + Y 36 0. In any case, the productexp(tX) exp(tY) belongs to the subgroup H, and for sufficiently small t wehave, by Lemma 1,

exp(tX) exp(tY) = exp(t(X +Y) + 0(t2)).

Then Z(t) := O(t2)/t apparently converges to zero for t - 0, and we canrewrite the preceding equation as

exp(tX) exp(tY) = exp (t(X +Y + Z(t))) E H.Choose a sequence of positive numbers that converges to zero, t - 0, anddefine Xn := tn(X + Y + Z(tn)). Each term exp(XX) lies in H, and

--:in = X+Y+Z(t,)=

X+Ylim

lim

11X-11 n-- IIX + Y + Z(t,a)[I Fix -+Y11Obviously, we have Xn # 0 and Xn -' 0. Thus, Lemma 3 applies, and wecan conclude that for every t E R

exp(tIIX+YII

is an element of H. Hence X + Y belongs to Fj.

Suppose that H is, in addition, a closed subgroup of G, and let h be definedas in the preceding lemma. We then choose any linear complement h' of hin 9,

9 = h+h'.Lemma 5. There exists a neighborhood V' C 4' of 0 such that, for everyX' 34 0 in V', the element exp(X') does not belong to H.


Proof. If the assertion were false, there would exist a sequence X;, E h'converging to zero and satisfying exp(X,,) E H. Now consider the compactset

K :_ {X' E h': 1 < JJX'II < 2}

and choose natural numbers p,, such that p ,,X,, E K. Since K is compact,we may assume that the sequence p,X;, converges to some 0 9& X' E K.Again, [exp(X;,)]P is an element of H and

pnX' _ X'lim

11p-Xnii 11X'6

Then Lemma 3 implies that X'/IIX'II E h, contradicting 0 0 X' E '. 0

Now we can turn to the main theorem of this section.

Theorem 7. Let G be a Lie group, and let H be a closed subgroup. Then:

(1) H is a submanifold of G and thus itself a Lie group.

(2) There exists precisely one differential structure on G/H such that(a) the projection 7r : G - G/H is smooth,(b) for every p E G/H there exist a neighborhood Wp C G/H of p

and a smooth map w : Wp G such that 7r o p = Idw,,,(c) the action of G on G/H defined by (g. kH) gkH is smooth.

Proof. It obviously suffices to show that there exists a neighborhood W C Gof e for which H n W is a submanifold (left translation is a diffeomorphismof G). As before, ddecompose the Lie algebra into g = 1) + h' and considerthe map corresponding to this decomposition,

4D : g = h + 4' ---i G, 4(X + X') = exp(X) exp(X').

In h' we choose a neighborhood V' C h' as in Lemma 5. as well as a subsetV C h so small that the exponential map still is a diffeomorphism on V +V'.The image W of V + V' under -ID is an open neighborhood of e E G, and

H n W =

by the definition of b and Lemma 5. The set H fl W is thus parametrizedby the chart (V, -D Iv+(o}), and hence a submanifold of G.

Now we turn to the proof of the second assertion. Let 7r : G - G/H denotethe projection. We define a topology on G/H by the condition

A C G/H is open :a 7r-' (A) C G is open.

It is called the quotient topology on G/H, and it is designed to render themap 7r continuous. Endowed with this topology, G/H is a Hausdorff space(see Duistermaat/Koik, Lemma 1.11.3). To verify the properties a manifold


has to satisfy, consider the distinguished point xo := e H E G/H togetherwith the sets V, V' introduced in the first part of the proof. The map

' : V' - G/H, X' a(exp X') ,

is continuous and maps V' onto an open neighborhood U of xo. Moreover, V,is injective, since v(X') = O(Y') implies the existence of an element h E Hsuch that exp(X') = exp(Y') h. Hence

h = exp(X') exp(-Y') = b(0 + (X' - Y')) .

Thus h also belongs to the set W, which was defined as the image of V + Vunder C Since we already proved H fl W = 4(V + {0}), this impliesX' = Y'. In summary, the map 1/i : V' U is continuous and bijective.For an arbitrary point gH E G/H, consider the left translation by g E G onG/H.

L9 : G/H G/H, kH gkHand introduce a chart around gH E G/H by

U9H L9(U), 09H: V' - UgH, Z'gH := Lg o TV .

For two points gH and kH, the chart transition can be rewritten as follows,

kH o y9H = v-1 o Lk-1 o L9 0 V1 = exp-1 o(ir-I o 4-1 0 L9 0 a) o exp

= exp-1 oLk-1h oexp .

Therefore. as a superposition of smooth maps, the chart transition is alsosmooth. Hence we have proved that G/H is a differentiable manifold, theprojection ;r : G - G/H is smooth, and C acts smoothly from the left onG/H. It remains to show (b). For the distinguished coset p = .ro = e H,define y; for each x in U =: Wp by

cp(x) = exp(Vi-1(x)) = 7r-1(x).

For an arbitrary point p = gH one again uses the left translation L9. 0Definition 5. The action of a Lie group G on a manifold Al is calledtransitive if, for two arbitrarily given points x and y in Al. the one canalways be written as the image of the other under the action of G. i. e.,there exists a g E G such that y = g x. An equivalent formulation of thisrequirement is to say that M consists of a single G-orbit, G x = Al. Amanifold together with a transitive group action is also called a homogeneousspace.

Obviously, the left translation on the quotient Al = G/H is a transitivegroup action, and thus G/H is a homogeneous space. Theorem 7 can beapplied to show that some well-known matrix groups are Lie groups: Thefollowing groups apparently are closed subgroups of GL(n, K).

6.3. The Adjoint Representation 221

Example 8. The subgroup of GL(n, K) consisting of all matrices with de-terminant 1 is a Lie group, the special linear group,

SL(n,K) :_ {A E Mn(K) : det A = 1} .

Example 9. Let H {[. I =: h u, v E c } be the vector space ofV

111

Hamilton's quaternions with standard basis

1

E =110

11' 1 = LO oil' J Lof OJ, K = Lo i 01

and norm N(h) := uu + vv. The group of all quaternions with norm 1 isisomorphic to the Lie group

SU(2) := JA E GL(n, IC) : AAt = 12 and det(A) = 1).

Example 10. The preceding example can be generalized as follows. Theunitary group is embedded into the space of complex matrices as

U(n) :_ {A E AAt = 1n} .

The condition AAt = 1n immediately implies I det Al = 1, hence det A E Sl;the special unitary group is defined as the group of all unitary matrices Asatisfying det A = 1:

SU(n) := {A E U(n) : det A = 1} .

Example 11. The orthogonal group O(n, K) consists of the matrices A EM ,,(K) leaving the euclidean standard scalar product of Kn invariant,

(Ax, Ay) _ (x, y)

Realizing the scalar product as (x, y) = xty, we see that this condition isequivalent to AAt = ln. Hence we obtain

O(n, K) = JA E Mn(K) : AA' = 1n} .

Obviously, an orthogonal matrix has determinant + 1 or -1. The subgroupof all orthogonal matrices with determinant +1 is called the special orthog-onal group SO(n, K),

SO(n,K) = {AEMn(K): AAt=lnand detA=1}.

6.3. The Adjoint Representation

Definition 6. Let G be a Lie group with Lie algebra g, and let V be afinite-dimensional vector space.


(1) A representation of the Lie group C on V is a smooth group ho-momorphism e : G GL(V), i.e., a smooth map compatible withthe group structure,

e(g h) = e(g) e(h)(2) A representation of the Lie algebm g on V is a homomorphism of

Lie algebras, p : g - gl(V), i. e., a linear map compatible with thecommutator,

e([X, Yl) = [e(X ), e(Y)] = e(X) e(Y) - e(Y), e(X)Sometimes, V is then also called a G-module or a g-module, respectively.

Example 12. The trivial representation of a Lie group G is the grouphomomorphism that maps every element g E G to the neutral element inGL(V): p(g) = 1V; the trivial representation of g associates the zero mapwith every element X, o(X) = 0v.

Example 13. Matrix groups are defined by means of one of their represen-tations, often called the defining representation. In fact, we introduced thegroups GL(n, R), SL(n, R) and SO(n, R) in a way endowing them naturallywith a representation on R". A simple example illustrates that these matrixgroups and their Lie algebras have many more representations. The Lie al-gebra sl(2, R), for example, has representations in all dimensions: For everynatural number n, define e : s((2, R) - gl(n + 1, R) by

g(H) = diag(n, n - 2, ..., -(n - 2), -n),

0 1 00 2

n 0

e(E) e(F) = n-1 0

0 n-L 1 0

These matrices satisfy the commutator relations of sl(2, R),

(E), [e (H), e(F)l = -2e(F), [e(E), e(F)] = e(H),[e(H), e(E)1 = 2,oand hence form an (n + 1)-dimensional representation of sl(2, R). Propertiesthat cannot be expressed by the Lie bracket do not have to be preservedunder a representation: For example, we have E2 = 0, but e(E)2 0 0.Nevertheless, the property that g(E) is a nilpotent matrix is preserved.There is also a a representation of the Lie group SL(2, R) corresponding tothis representation of the Lie algebra; this will be the subject of Exercise 4.

Apart from left and right translation, there is a third remarkable action ofa given Lie group G on itself, the so-called conjugation action,

ag : G - G, a9(h) := ghg-' = L9R9-, h.


It is smooth and satisfies ag(e) = e, and in contrast to left and right trans-lation, it is far from being transitive. In the case G = GL(V), it decomposesthe invertible matrices precisely into their similarity classes. In addition,the relation ag(e) = e implies that its differential at e is a map from g to g,

d(ag)e: TG 5--- g ----+ TG 2--- 9,

which is obviously invertible, since d(Lg)e and d(Rg-i)e are invertible. Wedefine the adjoint representation of G on g by

Ad : G ---+ GL(g), Ad(g) = d(ag)e E GL(g).

Before we verify that this actually is a representation, recall the definitionof the center of a group. It consists of those elements which commute withall the others:

ZG = {gEG: gh=hgdhEG}.

Theorem 8. The map Ad : G - GL(g) is a representation of G on thevector space g. The center ZG of G is contained in its kernel,

ZG C ker Ad,

and equality holds if and only if G is connected.

Proof. First we check the homomorphism property:

Ad(gh) = d(LghRgh )e = d(LgLhRh-, Rg-i)e = d(agah)e= d(ag)ed(ah)e = Ad(g)Ad(h).

Let z belong to the center Z. Then aZ = IdG, and hence Ad(z) = IdGL(9),i.e., z is in the kernel of Ad. Now suppose that G is connected and thatAd(g) = Ide. Since ag : G - G is a group homomorphism, the mapt ' -+ ag (exp tX) is a one-parameter subgroup for every X E g, and, byTheorem 5, there exists an element Y E g such that

exp(tY) = ag(exptX).

Differentiating this equation with respect to t, we obtain

Y(e) = dt (ag(exp tX )) Ie=o = Ad(g) (X (e)) = X (e) ,

which proves X = Y. For the one-parameter group defined above this meansthat

exp(tX) = ag(exptX)

for all t E R and X E 9. The exponential map is a local diffeomorphismg - G; hence ag = IdG on an open neighborhood W of e. For a connected


Lie group G this implies ay = IdG, since G can be represented as the unionof all powers of W (with respect to the group products),

00

G = UW'.

As a9 = Idc is equivalent to g E ZZ, everything is proved. 0

The differential of the adjoint representation of G (in the sense of Definition4) is a representation of the Lie algebra g which can now be expressed bythe commutator.

Theorem 9. The differential ad := Ad. : g -. gl(g) of the adjointrepresentation is a homomorphism of Lie algebras determined by the formula

ad(X)(Y) = [X, YJ.

Proof. By the definition of the differential, we have

Ad(exptX) = exp(tAd.(X)) = 1 + tAd.(X) + ... ;

hence

Ad.(X)(Y) =li.o

Ad(exptX)(Y) - Y

The flow corresponding to the vector field -X is (bt = R P(_tX), since

d4ie(e)

LO

dexp(-tX) I Xdtdt t=o

Applied to a left-invariant vector field Y, however, its differential coincideswith Ad(exptX)(Y),

Ad(exptX)(Y) = dL P(tx)dR P(-ex)(Y) = dR p(_tx)(Y) = d4it(Y).

Thus the original identity can be rewritten as

Ad.(X)(Y) = li o `ht(Y) - Y

The right-hand side is precisely the definition of the commutator [Y, -X] _[X, Y]. O

This representation will also be called the adjoint representation (this timeof the Lie algebra g). In case of doubt, the context has to decide whetherthe representation of the Lie group or that of its Lie algebra is meant.

Remark. The definition of the differential immediately implies the identity

Ad(expX) = exp(adX).


This has to be understood as an identity of operators. Applied to an elementY, it means

exp(X) Y exp(-X) = 1 + ad(X)(Y) +ad(X)3i3(Y)

+ ...

= 1 + [X, Y1 + [X, [2I Y11 + [X, [X 3'XI Y111 + ... .

Example 14. Let g be the three-dimensional Lie algebra which is abstractlydefined by the following commutator relations for a basis el, e2, e3:

[el, e21 = e3, [e2, e31 = el, [e3, ei1 = e2.

The representing matrices of the adjoint representation with respect to thisbasis can be computed from them. For the operator ad(ei ), we obtain

0

0

rO

l i

ad(ei) e2 e3 = 0 1 e2 Ll e2e3 -e2 -1 0 e3 e3

and similarly for the other two operators,

rol

0-1 0 10L2 := ad(e2) 0 0 , L3 := ad(e3) _ 1 0 0

0 0 0 0 0

Let us look, on the other hand, more closely at the orthogonal group O(n, R).It was defined as the set of matrices satisfying f (A) = AAt - 1, = 0. Thedifferential of this map at the point X is

df(A)x = AXt+XAt,

and hence, according to Theorem 5 in §3.2, the Lie algebra of O(n, R) is

o(n, R) = Te O(n, R) = {A E M,(R) : A + At = 01.

The Lie algebra of the orthogonal group consists precisely of the skew-symmetric matrices, which for n = 3 is apparently spanned by Li, L2 and L3.This proves that the three-dimensional defining representation of o(3, R) isisomorphic to the adjoint representation. In higher dimensions, this fact nolonger holds as a simple dimensional consideration shows: A skew-symmetricmatrix has exactly as many degrees of freedom as entries above the diagonal.Therefore,

dim o(n, R) =22 - n = n(n2 1)

and this is equal to n only for n = 3.


Exercises

1. The exponential map exp : si(2, R) SL(2, R) is not surjective. Hint:What are the values that tr exp(A) can attain for A E al(2, R)?

2. Hamilton's quaternions H (Example 9) form not only a vector space,but also a (non-abelian) division algebra, i. e., an associative algebra inwhich each non-trivial element is invertible. Prove that the standard basisE, I, J, K of Hamilton's quaternions obeys the following algebra relations:

I.K=-J,and compute the inverse of the quaternion h = 0.

3. Identify the quaternions of trace 0,

Ho = {xi . I + x2 J + x3 K j xi, x2, x3 E R},

with the 3-dimensional euclidean space R3. Prove that, for every U E SU(2),the map

Ho - Ho, x UxU-1,

defines a special orthogonal transformation of R3. The resulting map eSU(2) - SO(3,R) is a representation which is not injective ("faithful").

4. The defining representation of SL(2, R) on R2 is the usual matrix actionon vectors,

g VIc d] [y] [cx + dy]

Let V,, be the (n + 1)-dimensional vector space of homogeneous polynomialsof degree n in the variables x and y. Define an action of g E SL(2, R) on thepolynomial p E V,, by

P(g) ' P ([X]) = P\g_1 [;]).Prove that B is a representation of SL(2, R).

5. Let G be a Lie group, and let H be a discrete, normal subgroup of G.Prove that H is necessarily contained in the center of G.

Exercises 227

6. Let (p, V) be a representation of the group G. A subspace W of therepresentation space V is called invariant if. for every g E G. the relationp(g)W C W holds. For trivial reasons, the subspaces W = V and W = {0}are invariant: if the representation has no further invariant subspaces, it iscalled irreducible. Consider the following two-dimensional representation ofthe additive group R:

r 1

LOW =rI tlL0 11

Prove that this representation is not irreducible. Does the invariant subspacehave an invariant complement?

7. By Theorem 6. the differential of a representation (p. V) of the Lie groupG is a representation (Lo., V) of its Lie algebra g. The tensor product of tworepresentations (p, V) and (µ, W) of G is defined by

(p ®Fr)(9)(v $ w) := p(9)v (9 p(g)w, g E G. V E V. W E W .

Prove that this determines a representation of G on V O W with differential

(e® i),(X)(v ®w) := p,(X )v ®w + v ®p.(X)w, X E g.

8. In order to describe the hyperbolic plane as a homogeneous space. it isuseful to introduce a new model for it, the open unit disc.

a) Let D = {z E C I lzj < 1} be the open unit disc with metric

9 = (1_1212)2 I0 i`

I.Show that the Cayley transform.

c: c(z) = -i`+i=:x+iy,

z - iis an isometry between D with the metric above and the upper half-plane?{2 with the fourfold of the hyperbolic metric.

b) Let the Lie group

SU(1.1) :_ {lb b : 1a12 - 1b12 = 1}

act on D via the formulaa b _ az+bb a

z

az+bProve that this action is transitive, and that the isotropy group of zero,

Go := {g E SU(1,1) : g 0 = 0},is isomorphic to SO(2,IR). Hence D 5 SU(1,1)/SO(2,1R).

Chapter 7

Symplectic Geometryand Mechanics

7.1. Symplectic Manifolds

Riemannian geometry is the geometry of a symmetric, bilinear form depend-ing on the point of a manifold. The curvature is a measure of how far twosymmetric bilinear forms differ locally. Contrary to this, symplectic geom-etry is that of an antisymmetric bilinear form depending on the point ofa manifold-hence the geometry of a 2-form w. It turns out that all sym-plectic manifolds are locally equivalent: there cannot be any concept similarto curvature in the sense of Riemannian geometry. Symplectic structuresdiffer, if at all, only globally. Historically, the formulation of mechanics inthe sense of Hamilton led to symplectic geometry, hence its essential role inmodern mathematical physics.

Definition 1. A symplectic manifold is a pair (1112., w) consisting of amanifold 1112ni of even dimension together with a closed non-degenerate 2-form w,

d w = 0 and A

called the symplectic form or symplectic structure.

By Theorem 16 in §3.4, every symplectic manifold is orientable. The volumeform is understood to be the 2m-form

-(--1)/2dM2m = (-1) ' win .

m!

229

230 7. Symplectic Geometry and Mechanics

Example 1. In 1R2"' with coordinates {q1, ..., q,", pi, ..., p,,,), the formulam

E dpi A dqii=1

defines a symplectic form with highest power

w"' = m! - (-1)-(--1)/2 - dpl A ... A dpm A dq1 A ... A dq,,, .

The volume form in the sense of symplectic geometry is the ordinary volumeform of ]R2i'. This symplectic structure is called the canonical symplecticstructure.

Example 2. Define a 1-form 0-the so-called Liouville form--in the cotan-gent bundle T'X"' of an arbitrary m-dimensional manifold as follows: LetV E T,, (T' X ') be a tangent vector at q E T' X' and represent it by a curveV : (-e, e) - T* X' such that

V(0) = q, V(0) = V.

Project this curve first by means of the projection 1f : T'X"' - X"' to themanifold, and, after that, apply the 1-form q to the tangent vector of theprojected curve:

9(V) := n d(T ° Vd (t)) I=o

The 2-form w := dO is a symplectic structure on T'X'. Any system,{q1, ... , qm }, of coordinates in X' determines-representing a 1-form qas q = >2 pi dqi-coordinates {qt, ... , 9m, pi, ... , p,n } in TX". By thedefinition of the Liouville form 0, we have

m

0 = pi dqi

and the 2-formm

w=d9=EdpiAdgii=1

is non-degenerate. In particular, the (co-)tangent bundle of every manifoldis an orientable manifold (see Exercise 9 in Chapter 3).

Further examples of symplectic manifolds arise as the orbits of the coadjointrepresentation of a Lie group G. Starting from the adjoint representation,Ad : G GL(g), of the group G and passing to the dual of the linearoperator, Ad'(g) := (Ad(g-1))* : g' - g', we obtain a representation

Ad` : G -b GL(g')

7.1. Symplectic Manifolds 231

of the group G in the dual space g' of the vector space g. Through eachfunctional F E g' passes an orbit

01(F) := {Ad*(g)F : g E G},

on which the group G acts transitively. The isotropy group

GF := {g E G : Ad'(g)F = F)is a closed subgroup of G, and 0 *(F) is diffeomorphic to the homogeneousspace G/GF. Its Lie algebra can be characterized by a similar condition:

Theorem 1. The Lie algebra OF C g of the isotropy group GF C G is equalto

OF = {XEg :F([X,Y])=0 for all YEg}.

Proof. Suppose that F([X,YJ) = 0 holds for all elements Y E g. Thenfrom

(Ad*(exp(t X))F)(Y) = F(Ad(exp(- t X))Y) = F(exp(- t ad(X))Y)z

= F(Y - twe immediately obtain Ad*(exp(t X))F = F. The one-parameter groupexp(t X) is a subgroup of GF, and hence its tangent X belongs to the Liealgebra OF. This proves one inclusion, the converse is proved analogously.

0

If a Lie group G acts smoothly on a manifold Mm, we can associate witheach element X E g of the Lie algebra the unique vector field k on Mmwhose integral curves coincide with the trajectories of the one-parametertransformation group exp(t X):

dX(x) _

The vector field X is called the fundamental vector field corresponding tothe element X E g of the Lie algebra. If G acts transitively on Mm, everytangent vector V E TXM'" at a fixed point x E Mm can be realized by afundamental vector field.

This general construction will now be applied to an orbit O' of the coadjointrepresentation. First, realize a given vector V E TFG' as the value of afundamental vector field, f(F) = V. For a further element Y E g of theLie algebra such that k(F) = V, the equality (X-- Y)(F) = 0 immediatelyimplies

0.e=0


Theorem 1 implies X - Y E OF, and the resulting map is injective,

9/9F3X- X(F)ETFO'.For dimensional reasons, it is bijective: the tangent space TFO* to an orbit0* at F E O' can be identified with the vector space 9/9F. We now definea symplectic structure wo on each orbit O' C g'.

Definition 2. Let V, W E TFO' be two tangent vectors to the orbit atF E O', and choose elements X, Y E g with f ((F) = V, Y(F) = W. Thevalue of the Kirillov form wo on the vectors V, W is determined by theformula

wo (V, W) := F([X, Y])

Theorem 2. The pair (O', is a symplectic manifold, and the 2 formwo. is G-invariant.

Proof. Note first that the 2-form wo is uniquely defined. If the elementsX, X1 E g realize the vector V at F, then the difference X - X1 belongs tothe Lie algebra OF, and Theorem 1 implies

F([X,Y1) = F([X -X1,Y])+F([X1,Y]) = F([X1,Y]).Moreover, wo is a non-degenerate 2-form. If, in fact, wo (V, W) = 0 forevery tangent vector W E TFO', then we obtain F([X,Y]) = 0 for allelements Y E g. By Theorem 1, X lies in the Lie algebra OF, and henceV = f(F) = 0. It remains to show that wo is a closed form. For twoelements X, Y E 9, the function wo (X, k) : O' R is determined by theformula

F([X,YI)Differentiate this relation in the direction of a third fundamental vector field:

2(wo.(X,Y))(F) = d [Ad'(exp(-t.Z)F)[X,Y]]It_o

= dtF(Ad(exp(tZ))([X,Y]))It=o = F([Z, [X,YII)

Then the expression for the exterior derivative dwo of the 2-form vanishesidentically:

dwo. (X, Y, Z) = X (wo (Y, Z)) - Y(wo (X, Z)) + Z(wo (X, f))-wo.([Y,Z1,X),

since it reduces to the Jacobi identity of the Lie algebra g. 0Corollary 1. Each orbit O' C g' of the coadjoint representation of a Liegroup is a manifold of even dimension.

7.1. Sylnplectic Manifolds 233

Example 3. The affine group of R has the matrix representation

G = a>0,bER}{ [ 10 1

with Lie algebra

=1[Y l :x ER}9 0 0

The computation

.,yJ11

rAd

[01] [0 0] [0 1] [0

0]'[100a -b/a 1 _ 10 ay0bx1

implies that g has one-dimensional orbits. To determine the orbits of thecoadjoint representation, write any element of g' as a pair (a, 0) of realnumbers, whose evaluation on the element (x, y) E g is

((a, /3), (x, y)) = ax + /3y .

By definition, the group element g = 10 1] acts as follows:

(Ad* (g-1)(a, 3), (x, y)) = ((a, R), Ad(g)(x, y)) = ((a, Q), (x, ay - bx))= ax + /3(ay - bx) = ((a - fib, /3a), (x, y)) .

Summarizing, we have Ad*(g-1)(a,0) = (a - f3b,,Oa), and hence for 3 96 0the coadjoint orbit through (a,,3) is two-dimensional. This example showsthat the adjoint and the coadjoint representation of a group G are, in general,not equivalent.

After having discussed examples of symplectic manifolds, we now want tointroduce the symplectic gradient, which is the analogue of the gradient ofa function on a Riemannian manifold. In this situation, we will make useof the fact that the non-degenerate 2-form w also provides a linear bijectionbetween the tangent bundle TA12m and the cotangent bundle T* M2",.

Definition 3. Let H : M2, IR be a smooth function on a symplecticmanifold. The symplectic gradient s-grad(H) is the vector field on M2indefined by

w(V,s-grad(H)) := dH(V).

Example 4. Let {q1, ..., q,,,, p1i ... , pm } be coordinates on M2'" such thatthe symplectic form w can be written as w = E dpi A dq;. Then

s-grad(H) = (aH 0 aH a

This formula immediately follows from the equation defining the symplecticgradient. A curve -y(t) in the symplectic manifold M2rn, represented in the


fixed coordinates y(t) = {q1(t), .... q,n(t), pl (t), ... ,p,n(t)}, is thus an inte-gral curve of the vector field s-grad(H) if and only if the so-called Hamiltonequations hold:

aH aH9i = and Pi=--

api aqi

Theorem 3 (Liouville's Theorem). Let H be a function on a symplecticmanifold (M2m, w), and suppose that s-grad(H) is a complete vector fieldwith flow 4Dt : M2m , M2m. Then:

(1) The Lie derivative of w vanishes,

Gs-g,ad(H)(w) = 0.

(2) The flow 't preserves the symplectic volume,

JdM2m =

Jdm2m.

Proof. From dw = 0 and Theorem 32 in §3.9, we conclude that

Gggrad(H)(w) = d(s-grad(H) J w) = - dd H = 0.

Hence the diffeomorphisms 4)t : A f2m ,11m do not alter the symplecticstructure, i.e., 4 (w) = w, and so both assertions are proved. 0

The existence of this invariant measure has consequences for the dynamicsof symplectic gradient fields.

Theorem 4 (Poincare's Return Theorem). Let (M2,, w) be a symplecticmanifold of finite volume, and let 4bt : M2m _ hf2m be the flow of thesymplectic gradient of a smooth function H. For any set A C M2m ofpositive measure, the set

B = {x E A : tn(x) 0 A for all n = 1, 2, ...}

has measure zero.

Proof. Note first that the intersections 4>_n(B) fl B are empty. Any pointx E fl B would be a point x E B such that 4n(x) E B C A,contradicting the definition of the set B. This immediately also implies thatthe intersections ' _n(B) fl 4'_m(B) are empty for n m, and, from theinvariance of the measure, we obtain

x oc

Evol(B) = Evol((Dn(B)) < Vol(M2m) < x,n=1 n=1

i. e., the measure of the set B has to vanish. 0This result has several famous generalizations, as the only example of whichwe quote Birkhoff's ergodicity theorem (without proof).

?.1. Symplectic Manifolds 235

Theorem 5 (Birkhoff's Ergodicity Theorem). Let f be an integrable func-tion on the symplectic manifold (M2,, w), and let Ot be the flow of a sym-plectic gradient. Then the following limit exists almost everywhere:

lim 1J

t f o .0,(x) =: f` (x) .t o

Furthermore, the function f' is also integrable, and its integral coincideswith that of f. Lastly, f* is invariant under the flow fit.

Definition 4. The Poisson bracket of two functions f and g on a symplecticmanifold is the function

{f,9} = w(s-grad(f),s-grad(9)) = dg(s-grad(f)) = -df(s-grad(g)).

Example 5. In the {q, p}-coordinates, we have

{f, 9} _ m Of a9 _ Of a9i=1 \ apt 5q1 aq1 Opi

In the next theorem, we summarize the properties of the Poisson bracket.

Theorem 6. The ring C'°(M2,) endowed with the Poisson bracket is aLie-Poisson algebra:

(1) for constants cl, c2 E R;

(2) { f, g} = -{g, f };

(3) { f, {g, h}} + {g, {h, f }} + {h, { f,g}} = 0 (Jacobi identity);

(4) {f,g-h} =g {f,h}+h- {f - g};

(5) s-grad({f, 9}) = [s-grad(f ), s-grrd(9)J.

Proof. The two first identities result immediately from the definition ofthe Poisson bracket, and the fourth follows from d(g - h) = g - dh + h dg.We prove (5). We insert the vector fields V := s-grad(f ), W := s-grad(g)together with an additional vector field Y into the equation defining the3-form dw,

0 = dw(V,W,Y) = V(w(W, Y)) - W(w(V, Y)) + Y(w(V, W))- w([V, W], Y) + w([V, YJ, W) - w([W, Y), V) .

Applying the definition of the symplectic gradient as well as that of thePoisson bracket, we can rewrite this equation as

0 = -V (Y(9)) + W (Y (f )) + Y({ f, 9}) - w([V, W], Y)

+ [V,Y)(9)-[W,Yj(f) = -Y({f,9})+w(Y,[V,WI)


Hence s-grad({ f, g}) = [V, W] = [s-grad(f ), s-grad(g)J. The Jacobi identityis a consequence of formula (5). In fact, we obtain

{f. {g,h}}+{g.{h, f}}+ {h.{f,g}}= s-grad(f)(s-grad(g)(h)) - s-grad(g)(s-grad(f)(h)) -s-grad({f,g})(h)= [s-grad(f ), s-grad(g)] (h) - [s-grad(f ), s-grad(g)] (h) = 0.

A Hamiltonian system consists of a symplectic manifold (111211, W, H) to-gether with a function H. The integration of the corresponding Hamil-ton equation relies on determining the integral curves of the vector fields-grad(H). For this, there exists an analogous notion of first integrals as inthe Riemannian case.

Definition 5. A function f : 1112" -. R is called a first integral of theHamilton function if it is constant on each integral curve of the vector fields-grad(H).

Theorem 7.

(1) A function f is a first integral of the Hamilton function H if andonly if its Poisson bracket with H vanishes,

{ f, H} = 0.

(2) The set of all first integrals of a Hamilton function is a Lie-Poissonalgebra.

Proof. We compute the derivative of a function f along an integral curvey(t) of s-grad(H):

dt f o -y(t) = df (j(t)) = df (s-grad(H)) = {H, f } .

This implies the first assertion. The second follows from the Jacobi identityfor the Poisson bracket.

7.2. The Darboux Theorem

The Darboux theorem states that all symplectic manifolds are locally equiv-alent.

Theorem 8 (Darboux Theorem). Near each point x E 1112n of a symplecticmanifold (M2i/.w), there exists a chart h : U C 1112m - R2m in which thesymplectic form w is the pullback of the usual symplectic form.

w = h* dpi n dqi)e=1

7.2. The Darboux Theorem 237

Coordinates with these properties are called symplectic (canonical) coordi-nates.

Proof. In the cotangent space TTM2," to the manifold at the point x EM2",, we choose a basis al, ... , o , ILl...... m in which the symplecticform at this point is represented in normal form,

Iii

w(x) = E ai A pi .

i=1

Consider, moreover, a chart (D : V - 1R 2»i around x such that

4t(x) = 0 and w(x) = 4D` (dPAdQi(0))

Denoting the corresponding symplectic form on V by

Wi := fi' I dpi A dqi I ,

i=1

we see that there exists a neighborhood U C V of x such that for all param-eters t E [0,1] the form

wt :_ (1 - t)w + twl

is a symplectic structure on U. In fact, dwt = 0, and since wt(x) = w(x) forall t, a compactness argument shows that all the forms wt (t E [0,11) do notdegenerate at the same time in a neighborhood of x. Since d(wl - wo) = 0,Poincare's lemma shows the existence of a 1-forma such that the difference

w, -wo = da

is the exterior derivative of this 1-form. By subtracting locally, if necessary,a 1-form with constant coefficients from a, we may assume that a vanishesat the point x, a(x) = 0. Dualizing a by means of the symplectic forms wt,we obtain a family Wt of vector fields on U parametrized by t,

wt W, Wt) = a(V) .

Let cp(y, t) E Mgr" be the solution of the (non-autonomous) differentialequation

Ve(t) = Wt('p(t)), V(0) = yAll the vector fields W1(x) 0 vanish at the point x, and the solutioncorresponding to the initial condition x is constant, p(x, t) - x. Hencethere exists a neighborhood U1 C U of x such that, for every initial conditiony E U1, the corresponding solution p(y, t) is at least defined in the interval


[0, 1]. Let t : Ul -+ A12°' be the corresponding map. The formula for theLie derivative of a differential form (Theorem 32, §3.9) implies

dt 4 (wt) _ 'pi (L) + 0i ('Ca , ,a (Wt)) = V P1 - w) + Vi (Gww, ((WO)

= ipi (wi - w + d(Wt J wt) + W1 J dwt) = tipi (wt - w - da) = 0.

Thus Spi (wl) = pp(w) = w, and 4 o cpl is the chart we were looking for,

(DoVI)" Edpindyi w. O

7.3. First Integrals and the Moment Map

As in the Riemannian case, some first integrals can be derived from symme-try considerations. The isometries (which are not available on symplecticmanifolds) giving rise to these first integrals will be replaced by symplecticdiffeomorphisms which, however, satisfy a compatibilty condition with re-spect to the Hamilton function under consideration. We will describe thisin detail in the case of an exact symplectic manifold, i. e. , a symplectic man-ifold whose symplectic form is the exterior derivative of a 1-form, w = d9.Suppose that a Lie group G acts from the left on M2" in such a way thateach diffeomorphism 1g : M2i' -+ M2ni leaves the form 9 invariant,

l9(0) = 9.

These diffeomorphisms 1g are then symplectic, i.e., they preserve the sym-plectic structure w. Now let X E g be an element of the Lie algebra ofthe group G, and let k be the fundamental vector field corresponding toX under the G-action on M2in. The evaluation of the 1-form 9 on X is afunction,

.6(X) := 0(.k).This construction determines a linear Map '1 : g _ Coo(M2,n) from the Liealgebra g to the space of functions COD(M2in) on the symplectic manifold.Its properties are the subject of the following symplectic variant of Noether'stheorem.

Theorem 9 (Noether's Theorem).

(1) $ : g - C, (M2,) is a homomorphism of Lie algebras,

'DQX, Y]) = (4'(X ), CY)} .(2) s-grado4' corresponds to the transition to fundamental vector fields,

s-grad(qD(X)) = f(.

7.3. First Integrals and the Moment Map 239

(3) If the Hamilton function H is G-invariant, then is a firstintegral of H,

0.

Proof. Fix an element X E g in the Lie algebra and consider the one-parameter group of diffeomorphisms corresponding to the group elementsexp(-t X). Its generating vector field is the fundamental vector field X.The relation l9 (0) = 0 implies that the Lie derivative of 0 with respect to Xvanishes,

0 = cX(0) = X-jd0+d(XJ0) = X_jw+d(-t(X)).Thus, for every vector field V, we have the equation

-w(X,V) = V(4(X)) = w(V,s-grad(4'(X)))

as well as X = s-grad(4(X)). Using this formula, we compute the difference

{4(X),4(Y)} - .0([X, Y]) = w(X,Y) - 4,([X, Y])= dO(X,Y) - 4,([X, Y])

= X(4(Y)) -Y(,t(X)) -24([X,Y])= 2({'F(X), 4'(Y)} - -NX,Y])) ,

and this yields

{4(X), F(Y)} = C[X, Y]) .If, finally, the Hamilton function is G-invariant, we obtain

{H,fi(X)} = -X(H) = 0,i.e., 4(X) is a first integral. 0

The elements of the Lie algebra g provide first integrals for every G-invariantHamilton function. These first integrals can be combined into a singlevector-valued first integral by passing to the dual space g'.

Definition 6. The moment map of a Hamiltonian system with symmetrygroup G is the map IF : M2m -+ g' from the symplectic manifold to the dualof the Lie algebra defined by

IF(m)(X) CX)(m)Theorem 10.

(1) The map %P is Ad'-equivariant, i. e., the following diagram com-mutes:

M2m 19 M2.


(2) %V is a first integral of H.

Proof. The fundamental vector field X of a G-action has the followinginvariance property:

dX (1,(x)) = dt [exp(-tX) g x] Jt_o

_ dd lg

dt[exp(-t Ad(g-1)X) xJ It-0

= dlg(Ad(g-1)X(x)).

The 1-form 0 is G-invariant by assumption, and from this we obtainN N'P(lgx)X = 8(X(lg.x)) = B(Ad(g-1)X(x)) = '(x)(Ad(g-1)X). 0

In Exercise 11, we discuss the case M'' = T'R3 with symmetry groupSO(3, R) and its usual representation on R3. In particular, it is shown thatthe moment map P : T`R3 --+ so(3, R) = R3 coincides with classical angularmomentum, hence justifying its name. Closely related to this situation isthe following example:

Example 6. Consider the 2-dimensional representation of G = SL(2, R) onM2 = V = R2. Its cotangent bundle is T'M = V x V' ^' V x V, sincethe representation V is self-dual. A group element g E SL(2, R) acts on anelement (p, q) E V x V of the cotangent bundle by

g - (p, q) = (gR gq)-We call two elements (p, q) and (p', q') equivalent if they lie in the sameG-orbit, (p, q) - (p', q'). Let p = (pl, p2) and q = (ql, q2) be the componentsof the vectors p and q, respectively. One easily computes that the momentmap is given by

V x V - + s1(2, R), (q, p) -- 2 (g1P2 + g2P1) 1 -gipi[ q2P2 - 2(g1p2 + g2Pl)

In particular, this map is equivariant with respect to the adjoint action ofSL(2, R) on sl(2, R). The moment map is best studied by examining itsaction on SL(2, R)-orbits. For this, observe that the quantity

det(q, p) := det [qi pig2 P2

= g1P2 - 92P1

is SL(2, R)-invariant. It thus allows to parametrize the orbit space; we omitthe easy proof here'.

1For details on this SL(2,R)-action, we refer to §1.4. of the book by Hanspeter Kraft,Ceometnsche Afethoden in der Invariantentheorie. Vieweg, 1985.

7.4. Completely Integrable Hamiltonian Systems 241

Theorem 11.(1) If det(q,p) _: A 56 0, then

(q, p) - ( 0 A(1)1(0))(2) det(q,p) = 0 if and only if p and q are linearly dependent. In this

case, there exist infinitely many G-orbits, for which one can choosethe following representatives:

( 0 0 0 0 0 0(0)1(0)), ((1)1(0)), ((,U) W) with AERR.

Thus, the moment map acts on G-orbits as follows:

((01),(A0)) LA02

-A/2J' \(0)' 0// ~ [0 0 J

((0) ' (1)) ' (CO) ' Co)) [0 0

In particular, the generic orbits with parameter A 54 0 are mapped to semi-simple elements of the Lie algebra sl(2, R). Their orbits are 2-dimensionalclosed submanifolds of sl(2, J).

7.4. Completely Integrable Hamiltonian Systems

In this section we will make use of the following fact concerning the structureof discrete subgroups IF of the additive group IRA.

Theorem 12. Let t C ]RI be a discrete subgroup. Then there exist linearlyindependent vectors vl, ... , vk such that

k

r = m; vi : m; an integer

i. e., F is the lattice generated by the vectors vi, .... vk.

Proof. If I' # {0} is not trivial, we choose a vector ryl E r such thatI I7i I I 0 0 and consider the ball D" (0; I I7i I I) The intersection D" (0; I I71 I I) nris a compact and discrete subset of IR^, hence finite. Thus, on the straightline generated by 71i there exists a vector 7i E D° (0; 117, 1I) n r realizing theminimum of the distance to 0 E lR'. For this vector we have

Ht 7i n r = {m 7j : m an integer),

since any vector x 36 m7i belonging to the intersection (1R 7i) n r wouldhave to lie in one of the segments and then (m+1)7i -xwould be a vector on the line through 7t with a smaller distance to 0 than 7l*.If the group r contains only integer multiples of -y,*, the proof is completed.

242 7. Syrnplectic Geometry and Mechanics

Otherwise, there has to exist a vector 1'2 E r\{m - ryi : man integer}. Weproject ry2 orthogonally to the straight line passing through ry1 and denoteby Y2 the resulting vector. It lies in one of the half-closed segments y2 E[m - 'y , (m + 1) ryi ). Let E be the cylinder with axis [m' . (m + 1) - 7i)whose radius is equal to the distance from ry2 to the line through 'Y1. In thiscylinder, there are again only finitely many elements of the group r. Let ryebe the vector in r n E whose distance to the axis of the cylinder is minimaland which is not a multiple of ryi . Then we have

2

r n {R 7i ED R7;) mi ry, : mi an integer .

i=1

In fact, if there were a point x 0 miel + m2e2 in r n {Rryi ® R-y }, then xwould belong to the interior of a parallelogram in the {'y , 7s }-plane. Takingthe difference with a vertex of this parallelogram, we obtain a vector in rlying closer than -y; to the axis of the cylinder. Repeating this constructionfinitely many times proves the assertion. 0

Corollary 2. Let F C R" be a discrete subgroup. Then R"/r is diffeomor-phic to the product of a k-dimensional torus Tk with R"-k,

R"/r Tk x Rn-k

Theorem 13 (Arnold-Liouville Theorem). Let (M2m, w, H) be a Hamil-tonian system, and let fl = H, f2, ..., fm be m functions with the followingproperties:

(1) all functions fi are first integrals of H:

(2) the functions fi commute, { fi, f;} = 0;

(3) the differentials dfl, ..., df,,, are linearly independent at each point;

(4) the symplectic gradients s-grad(fi) are complete vector fields.

For a given point c = (cl, ... , c,,) E R'", we consider the level manifold

Al, = {x E M2m : fl(x) = C1, ..., fm(x) = C,n}.

Then:

a) The connected components of Al, are diffeomorphic to Tk x R'-k

b) The vector field s-grad(H) is tangent to Mc. In particular, eachintegral curve of this vector field is completely contained in one ofthe level manifolds.

c) If Mc is compact and connected, angle coordinates y91, ... , cp,,, canbe introduced in Mc -- T' so that the integral curves of s-grad(H)


are described by the system of differential equations

y , = vi, v; = constant.

Proof. Consider the flows 4i , ... , 4if :M2m M2, of the symplectic

gradients s-grad(fi). Since

0 = s-grad{fi,f3) = [s-grad(fi),s-grad(f3)1,

all these flows commute with one another (Theorem 36, §3.9). This deter-mines an action of the additive group R' on the manifold M2'":

(tl, ... , tm) x = .01 0 ...

The orbits of this R'-action coincide with the connected components of thelevel manifolds. In fact, since

0 = {ff,fi} = s-grad(fi)(f3),each function fi is constant on every orbit. Hence, the orbits of the R'"-action are contained in the level manifolds. On the other hand, both arem-dimensional submanifolds of M2in, since the differentials dfl, ..., dfm arelinearly independent. The isotropy group r(xo) = {t E Rm : t xo = xo}of a point xo E M2rn for the R'-action is discrete. Hence each componentof a level manifold is diffeomorphic to the product of a torus and euclideanspace:

Rm/r(xo) = T" x ][ -Assertions a) and b) are proved, so we turn to the remaining one. Supposethat a level manifold Mc is compact and connected. Choose a point xo E Mfand denote by v1, ... , v,1 E Rm a basis of the isotropy group r(xo). Repre-senting the basis vectors {v;) of the vector space I(tm as linear combinationsof the vectors in the standard basis e1, ... , em of R..

M

vi = 1: ajaeo ,a=1

we obtain a quadratic matrix A := (aid). Letm

r(m) _ {ni.ei : ni an integeri=1

denote the standard integral lattice in Rm. Thenm m

0: Rm/r(m) -. Rm/r(xo) _ - o Exi. ei) = E xi . Vi,i=1 i=1

defines a diffeomorphism. The inverse of this map,

4D-1 : MM -+ Rm/r(m) = S1 x ... x S',


as well as its components, 4b-1 = (WI, ..., cp"), lead to the angle coordinatesfor the level manifold M, In fact, if yl, ... , y' are the coordinates inR"'/r(xo) = MM determined by

1 m 1 M. VM,

then, by the construction of the R'-action on Af,

s-grad(fi) = y;

With respect to the gyom}-coordinates, this yields

s-grad(fi) =

maj,

a

a=1 awa

In particular. the symplectic gradient s-grad(H) is a vector field with con-stant coefficients on the torus MM = T', and the third assertion results bytaking v. := alb.

W e want to discuss more closely h o w the angle coordinates ( 1, ... , m) ofa compact and connected level manifold can be determined directly fromthe commuting first integrals. This will lead to an explicit algorithm forthe integration of a Hamiltonian system (M2m, w, H) provided with suffi-ciently many commuting first integrals. Because of this procedure, thesesystems are called completely integrable (or integrable by quadrature). De-note by wl (c), ... , ul,,, (c) the frame of 1-forms on Al, dual to the vector fieldss-grad(fl ), ..., s-grad(f,,,). The representation of the vector fields s-grad( f;)in terms of the vector fields 8/&pj immediately implies the following formulafor the differentials:

M

d'pi = aia wa(c)=1

Let ik be the closed curve in MM corresponding to the parameter valuesrpm=0. Then

m

aik = f dcpi = F, ai0 f wn(c)'Y k a=1 k

Hence, first the coefficients aid and then the angle coordinates can be com-puted directly from the first integrals. We summarize this in the form of analgorithm comprising five steps.

Step 1. Fix c = (cl, ... , c,,,) E IIt"' and let Al, be compact and connected.Choose a homology basis y', ... , y", for the first homology group H, (Al,: 7L).


Step 2. Compute the symplectic gradients s-grad(fi) of the first integralsf1=H,f2.....fm

Step 3. Determine the frame of 1-forms w1 (c), ..., wm(c) dual to the frameof vector fields s-grad(f l ), ... , s-grad (f.. ) on M.

Step 4. Compute the line integrals frk w0(c), and invert the resulting(m x m) matrix. This yields the matrix A = (aij(c)).

Step 5. Compute the angle coordinates (pi(c) on the level set MM from theequations

m

dvi = E ai0(c) wa(c), 1 < i < m.Q=1

This procedure computes the angle coordinates Vi(c) on one level manifoldMc. Note that, according to Step 5, these are only determined up to con-stants. As we vary the parameters c = (cl, ... , c,,,), the Cpl, ..., cp,,, becomefunctions on an open neighborhood of a level manifold Mc C M2m. Sincethe symplectic gradients are tangent to Mc, the Poisson bracket with theoriginal functions f1, . . . , fm is computed by

IVi,fjI = dpi(s-gradfj) = aij(fl, . . . , fm).

Moreover, it is a function exclusively depending on fl, . . . , fm. Similarly,we prove that the functions {Vi, cpj } are also constant on the level sets.

Lemma 1. The Poisson brackets {cpj,Vj} = bij(fl, ...,fm) are functionsdepending only on fl, ... , fm. In particular, they are constant on each levelset 't1c.

Proof. We compute the derivative of {vi, wj } with respect to the vectorfield s-grad(fk):

{{Vi,'j},fk} _ {Wj,fk},'Pi} - {{fk,'ci},'pj}_ -{aik(fl, ...,fm),'Pi} + {aik /lfl, ...,fm),pj}

m COajk m 49 }Oya

a_Ia=1

8ajk aaEk

fta

m

as -aj.)

Thus all the derivatives cpj} (1 < k < m) are constant on ev-ery torus T' = Afc. But then the Poisson bracket {Vi, oj} itself is constanton every level manifold Mc. 0


Now we alter the angle coordinates, which up to now were considered onlyon a single level manifold, by a suitable constant on adjacent level manifolds.The aim of choosing these constants of integration for the angle coordinatesis to obtain functions cpi commuting on M2n'.

Lemma 2. There exist functions Bl (yl, ... , B," (yl, ... , y') suchthat the angle coordinates

Bpi := Wi+Bi(fl,...,fm)commute on the symplectic manifold M2m, {(pi , Vj*) = 0.

Proof. Using the notation {cpi, f;} = aij and {Wi,co } = bij, we applythe Jacobi identity to the triples (cpi, ip,, fk) and (ipi, Wj, cpk), and take intoaccount the fact that the Poisson brackets { fi, fl) vanish. Thus we obtainthe relations

maaik Oa,ja;a- aia = 0,E

Inserting also the coefficients air of the inverse of the matrix A = (aid), weconsider the 2-form

m mIl :_ > bijai°a'p dy° A dyO.

i,j=1 a,p=1

The above relations say that fl is a closed 2-form. In fact, the first relationmeans that the 1-forms

m

of E a° dy°a=1

are closed, doi = 0. Hence we can choose coordinates z1, ... , z'" such thatof = dzi, and 1 becomes

mObi; ab;k 8bki 1E y u 'aka +

ft la' afa + d ajQ J = 0 .

M

fl = > bij dzi Adzj .

i,j=1

Since

Obi; _ mObi, ay°

Ozk 1 Oya 8zk8bijor aka

1or

a=the second relation then precisely expresses the vanishing of the exteriorderivative dfl = 0. The angle coordinates we set out to find are now takento have the form

m

Bpi Wi + aiaBaO=1


with functions Bl*, ..., Bm depending only on fl, ... , f,,,. Thenm

a;l_ L

aY

syvai. B.

MaB;

+ a (aiaaj3 - ajasid)a.8=1

Taking into account the first of the relations above, the condition0 turns out to be equivalent to

1: ".bij =a 33

(aiflaja - ai.aj,3)a.3=1

This system of differential equations can be reformulated as

aB M ma r3 - [: bijaiaajZ,

8ii - aya -i;=1

and, by Poincare's lemma, it has a solution, since the 2-form f) consideredabove is closed.

Thus we can determine the constants of integration occurring in the transi-tion from the differentials

m

d'pi(c) = E afa(r) wa(c)a=1

to the angle coordinates on the individual level manifolds in such a way thatthe functions Vi, defined in a neighborhood of a level manifold, commuteon M2'". Now we add so-called action coordinates J1, ..., J,, and therebybring the symplectic structure near a level manifold into normal form. TheHamilton function H = f, is itself constant on the level manifolds and onlydepends on the action variables, H = H(J1, ...,J,,,).

Theorem 14 (Action and Angle Coordinates). In a neighborhood of anycompact, connected level manifold M,: C At" of a Hamiltonian system de-termined by m commuting integrals fl, ... , fm, there exist angle coordinatesCpl, ... , v,,, and first integrals J1, .... J,,, such that

in

w = dindJi.i=1

In particular, this implies Vj} = 0 = {Ji, Jj} and {y'i, Jj} = bij.

Proof. First, we determine the angle coordinates near the compact, con-nected level manifold MM such that

J(pj, ypj} = 0 and {tpi, fj} = aij(fi, fm)


We look for the functions J1, ..., J,,, using the Ansatz J, = Ai(fl,and compute the Poisson bracket

m8Aj ai.OY.

The condition {,..1j} = 8ij leads to the system of equations

OA = aijayj

where a'j is the inverse of the matrix aij. By Poincare's lemma, the inte-grability condition is

8aij 8aik8yk = $yjj

On the other hand, we obtain from the Jacobi identity

0 = {`r'k,{iPi,.fj}}+{Vi.{fj, 7k}}+{.f .{Y"'k,Y^'i}},

and, taking into account {j,9i.k} = 0, this immediately yields

8aq = L 8ukjaka 8 a aip..E

Q=1 fta a=1 y

A simple computation shows that this relation is equivalent to the integra-bility condition for the coefficients aij of the inverse matrix. 0Example 7 (Two-dimensional Hamiltonian System). Consider in R2 withthe symplectic structure w = dp A dq a Hamilton function H(q, p) for whichthe level curves

(q, P) E 1R2 : H(q. p) = c}

are closed. The action variable J = J(H) is a function of H, and, togetherwith the angle coordinate , we have

dpAdq = dV AV =

Applying the Hodge operator * of R2, we see that dy: is proportional to the1-form *dH,

dVJ'(H) IIdHII2

* dH .

The integral of d,o over each level curve is an integer, which we take tobe equal to -1. This condition is called the classical Bohr-Sommerfeldcondition. The equation

J'(c) 1 * dHnor IIdHII2


uniquely determines the action variable J = J(H) in terms of the Hamiltonfunction H. Consider the domain bounded by the level curve A4,,

Q, = {(q, p) E R2: H(9, p) 5

C1.

The vector field W := grad(H)/Ilgrad(H)II2 satisfies W(H) - 1; hence itsflow +t maps the set flc onto tl +t. We compute the change of the area ofthe domain:

(J$2)d(vol(S2c)) =

tli ot t - J 2) = J Gbb'(dR2)

r

Jk st4 s24

.

Jd(W i dlt2) = W I dR2 = If4 IIdHII2 *

dH = X(c)i1c lo ,1

Thus the action variable J = J(H) can be interpreted as follows: J(c) isthe volume vol(Q,) of the domain bounded by the level curve hf, _ {(q, p) ER2 : H(q,p) = c}.

Example 8 (Spherical Pendulum). Consider spherical coordinates on thesphere S2\{N, S} with the north and south pole deleted,

h(4,) = (cos cp cos ti, sin yp cos 0 , sin g) .

The Riemannian metric of the sphere is then described by the matrix (seeExample 14 in §3.2)

_L

1 09 O cost'

Denote the coordinates in the cotangent bundle by (cp, o, pw, p v) and con-sider the Hamilton function

H = 2v+2H describes the motion of a pendulum of length one suspended in the centerof the sphere. The meaning of the angles tp and t'} can be seen in the pictureto follow. For simplicity, the gravitational constant was taken to be one.


Z

The variable (p does not explicitly occur in the Hamilton function; henceP:= p,, = Ocos2 iii is a first integral, {H, P} = 0. The Hamiltonian system(T*S2, H) is thus completely integrable. The level manifold

M2(ci,c2) :_ {pw,po) E T* S2 : P = c1, H = C2}is empty for negative values of the parameter c2, and it consists of thesouth pole remaining at rest in the case c2 = 0. Hence we suppose thatthe parameter is positive, c2 > 0. The equations describing the manifoldA12(cl,c2) are

C1 = pr,, c2 = 2 + 2 cs2 i + 1 + sin ?P.

The relation cl = 0 implies that cp has to be constant. In this case, thesecond equation of motion reduces to that of a planar pendulum, so we willhenceforth exclude this case. Depending upon the sign of c1, the functionis monotone increasing or decreasing, i. e., the pendulum does not change itsdirection of motion. Rewriting the second equation and setting z = sin 4i,we obtain

C- 2p=c2-1-z- 2(1 z2) U(z)

and see that the function U(z) thus defined cannot be negative. The limitingcase p = 0 corresponds to the pendulum moving in a fixed plane, henceon a meridian. To see where the function U(z) can be strictly positive, wemultiply it by the denominator 1- z2 and look for the zeroes of the resultingcubic polynomial,

c2V(z) = (1 - z2)U(z) = (c2 - 1 - z)(1 - z2) - 2 .


At the boundaries of the interval, V(±1) = -c/2 < 0 is negative, and at+oe the function V diverges to +oc. Hence one of the three zeroes has tolie above 1, and, since it cannot correspond to any angle 1;i, this zero hasno physical relevance. In the generic case, the other zeroes belong to theinterval (-1, 1); between them V, and hence also U, is positive. We concludethat V(z) has the qualitative behavior of the graph on the previous page.The mass point can only move between the two meridians corresponding tothe zeroes in the interval (-1, 1). The boundary values t ,'4'2 defined thisway are actually reached at the end of every up or down swing. Summarizing,

the level manifold can be parametrized by the two parameters 4z = V and0 = via

( P Cl! cos - 2 - 2sini ) .

It consists of 2 two-dimensional tori, where ' only takes values in (>G1, 021.The corresponding coordinate vector fields are

a a a a apti; aa = TT' a = a + o' ap ,

We express the symplectic gradients of the first integrals in the coordinatesof the level manifold:

P;, a a d P,2. a+ 1 + sin V)s-grad(H) = cost ;, a + p av - a (2 cost V, aplo

Pro a a= cost' ap +Pva +Pv a i "PV

pw a a_ 2 Zi app. + Pv a,.

s-grad(P) = a = ate, .


The dual forms w1(cl , c2) and w2(cl, c2) are thus

wl(Cl,C2) =1

de*, w2(cl,c2) = dw -C12

dip .

PO p o Cos '+G

We compute the periods with respect to the homology cycle 'rl which isparametrized in M2(c1i C2) by 't()'. The factor 2 is a consequence of thefact that the boundaries, y1,'Y2, of the interval correspond to the minimumand the maximum of the motion, whereas a cycle is meant to be a motionbetween two extremal points of the same kind:

rb1l = f wl = 2

J% b12 = f w2 = -2c1

J02 diP

oos2ti c>> Py 7i v PbLet, similarly, 72 denote the homology cycle determined by 0 < gyp` < 2Tr.Then

b21 = Jwi = 0, b22 = Jw2 = 27r .

The matrix occurring in the algorithm for computing the angle coordinatesis now easily calculated:

1 -j w2

[b21 b221

-l- [a21

a22] f7twl 2vfnwl0 1/27r

The resulting quotient of the basic frequencies is

V1 =all =_1f1-'2=.//

C1 112 di&

V2 a12 21r ,7r ol cost tai ' PW

This is nothing but the perihelion precession, i.e. the total variation of theangle V for a complete cycle:

fo fo,

O +aJ

dye = 2 J d` d1/b = 202

. dpi = 2c1 d =27r L1 .

'rI .1,1 d1 , I

COS2t' ' PG itIn general, the motion is quasi-periodic. The trajectory is closed if andonly if vl /v2 is rational; otherwise, it is dense on the torus. The transitionto action angle coordinates allows us to determine the physically relevantbasic frequencies of the system without having to explicitly perform theintegration. This accounts for their importance in astronomic perturbationtheory.

7.5. Formulations of Mechanics

Newton's equations describe the motion of a mechanical system under theimpact of a force. The latter is understood as a vector field depending uponposition and velocity, and is central for Newton's formulation of mechanics.

7.5. Formulations of Mechanics 253

During the 18th century, this view changed in that Lagrange considered theaction integral as the fundamental quantity for the description of dynam-ics. Newton as well as Lagrange formulated mechanics within the tangentbundle of configuration space. In the 19th century, by transition to thecotangent bundle, Hamilton succeeded in formulating the dynamics of me-chanical systems within the framework of symplectic geometry. The aimof this section is to explain these fundamental ideas of mechanics and therelated mathematical structures.

Newton (1643-1727) Lagrange (1736-1813) Hamilton (1805-1865)

Newtonian systems

i

Newtonian systems

with potential energy

Mathematical Contents:

Lagrangian systems

1

Hamiltonian systems

hyper-regular

Lagrangian systems

Legendre

transformations

Riemannian geometry Finsler geometry symplectic geometry

Formulation of Mechanics According to Newton

In Newtonian mechanics, the state of a mechanical system is described byfinitely many real parameters. This leads to the notion of configurationspace. which is a smooth and finite-dimensional manifold Mm. A motion ofthe mechanical system is a curve 'y : (a, b) - M' in configuration space.Its tangent-the velocity-is then a curve ' : (a, b) - TM' in the tangentbundle. and this space is called the phase space. According to Newton,the forces acting on the mechanical system are described by vector fieldsdepending only on position and velocity, that is, vector fields X on TM"'.However, not all vector fields are allowed, since a force can act only in space.


The force vector field X has to satisfy the condition

drr o X = Id.

Here 7r : TM" - Mm denotes the projection of the tangent bundle, anddir : TTM' --+ TM' is its differential. Summarizing, we arrive at thenotion of a Newtonian system.

Definition 7. An autonomous Newtonian system is a triple (Mm, g, X )consisting of a manifold M'", a Riemannian metric g, and a vector field onthe space TM'" such that

d7r o X = IdTMm .

The function T : TM'" -+ R defined by

T(v) := 2 g(v,v)

is called the kinetic energy.

Definition 8. A motion of the Newtonian system (M"', g, X) is understoodto be a curve y : (a, b) - M' in configuration space whose curve of tangentsry : (a, b) - TM'" is an integral curve of X,

'1'(t) = X MWThis is an invariant formulation of Newton's equation.

Example 9. Consider R with the coordinate x and identify TR = R2 withR2. Here the coordinates are denoted by {x, i}. The vector field

8 1 2 0X = wax+m(-k x-P)8i

on TR has the required projection property, and a curve x(t) in R is a motionin this Newtonian system if x(t) is a solution of the oscillator equation

ml(t) = -k2x(t) - pe(t) .

Example 10. Let (Mm,g) be a Riemannian manifold. We define a vec-tor field S : TM°1 - TTMm on its tangent bundle-the so-called geodesicspray-as follows: If v E TAM' is a tangent vector, then there exists pre-cisely one geodesic line y,,,,, : (-e, c) -> Mm such that

7'r,.(0) = X, % AO) = v.Consider its tangent curve, %,v : (-e, e) - TM'", and set

S(v) dt (%"(t)) t=o

The relation 7r o ryx,,, (t) = ry v (t) implies dir o S = IdTMm. Hence (M', g, S)is a Newtonian system, and the motions of this system are the geodesic lines


of the Riemannian manifold (Mm, g). In the coordinates {x', ii} of thetangent bundle, the geodesic spray is given by the formula

S = x'axi

- E r;k x'xA axi

i=1 i,j,k=1

a straightforward consequence of the system of differential equations de-scribing geodesic lines-see §5.7.

Many Newtonian systems have a potential energy. This is a smooth functionV : Mm --+ R defined on configuration space. The gradient grad(V) is avector field on Mm locally determined by

'" av agrad(V) - ` g'' (x) axi axji,j=1

In the sequel we will need, however, a different vector field, denoted bygrad(V). This will be a vector field on phase space. At the point v E TM'",it is defined by the following equation:

grad(V) (v) =ddt

(v + t - grad(V)(ir(v))) e-o-

In the manifold Mm, the curve v + t - grad(V)(a(v)) projects to the basepoint rr(v) E M' of the vector v E TM'. Hence the vector field grad(V)projects to zero under the differential dir,

d7r o grad(V) = 0,

and, for any potential energy V, the vector field X := S - grad(V) is anadmissible vector field on TM'" in the sense of Newtonian mechanics. Inlocal coordinates, we obtain the formula

in av aE e(x) axi aiji,j=1

Definition 9. A Newtonian system with potential energy is a triple (M, g, V)consisting of a Riemannian metric g and a potential energy V : M' -+ R.The corresponding force vector field is

X = S-grad(V).A motion in a Newtonian system with potential energy is defined to be acurve 7 : (a, b) Mm whose tangential curve (a, b) -+ TM'" is anintegral curve of X.

Definition 10. The energy of a Newtonian system (Mm, g, V) with poten-tial energy is the sum of the kinetic and the potential energy,

E:TM' R, E=T+Voir.


Theorem 15 (Energy Conservation for Newtonian Systems). Let (Mm, g, V)be a Newtonian system with potential energy, and let X = S - grad(V) bethe force vector field. Then

dE(X) = 0.

In particular, E(y(t)) is constant for every motion-y(t) of the system.

Proof. In local coordinates, the energy E and the force vector field aregiven by the formulas

I M

E = 9 E 9ij(x)xY +V(x),i,j=1

X = x' a ii - I'v (x) + 8i9ikW 82k.

i=1 k=1 i=1 IUsing the expression for the Christoffel symbols riki from §5.7,

,-ij2gkQ (x) (Lgii (x) + 8x (x) axQ (X))

we obtain dE(X) = X (E) = 0 by an elementary calculation. 0

Motions with large energy in a Newtonian system (M'", g, V) with potentialenergy are-up to a change of parametrization-geodesic lines with respectto a new Riemannian metric. This construction will lead to the Maupertuis-Jacobi principle. Suppose that the potential energy V : Mm - R is boundedfrom above by Eo,

sup{V(x): xEMm} < Eo.

Then g' = (Eo-V) -g is a Riemannian metric on the manifold Mm. Considera motion y : (a, b) - MI of the Newtonian system (MI, g, V) with energyEo,

Eo = 29('Y(t),'Y(t))+V(7(t))

Since Eo > V(y(t)), the tangent vector y(t) vanishes nowhere, and thefunction

s(t) := f r' (Eo - V(7(µ)))dpa

becomes a strictly monotone function a : (a, b) - (0, b* ). We invert thisfunction and thus view t E (a, b) as a function of the parameter s E (0, b*),t = t(s). Let the curve .y*(s) be the initial curve y in this new parametriza-tion.


Theorem 16 (Maupertuis-Jacobi principle). Let -y(t) be a motion of theNewtonian system (Mm, g, V) with energy E0. Then ry' (s) is a geodesic linein Mm with respect to the Riemannian metric g' = (Eo - V) g.

Proof. The Christoffel symbols r and 't of the metrics g and g' are, in1) Vlocal coordinates, related by the formula

k = k 1 1 _ 8V 8V 8V akrtj r'3 + 2 (Eo - V) ajk C7xi

-&k +02-a

9 9ij

We write the motion -y(t) _ (x' (t), ... ,x'(t)) in local coordinates. Then

dxk dsk dt _ 1 1 dxkds

_dt Ws- 72= (Eo - V) dt

d2xk _ 1 d2xk 1 dxk m 8V dxads2 2(Eo - V)2 dt2 2(Eo - V)3 dt 1 Ox- dt '

and, using the equation of motion,

d2xk m rk Ix' dx' mOV ka

dt2 - - ij dt dt - 8xa 9i,j=1 a=1

as well as the energy conditionm dxi dxjE gij dt dt

2(Eo - V),i,j=i

we obtain the claimed result:

d2xk m k dx' dx' 1 8V kaWS-2 + I'i ds ds

=-2(Eo - V)2 8xa9ij=1 a=1

1m 8V ka in dx' dx-I

+ `1(Ec, __V) 3 8xa9 go dt dt = 0.a=1 ij=1

Formulation of Mechanics According to Lagrange

The transition to Lagrangian mechanics proceeds by considering the La-grange function of a Newtonian system with potential energy.

Definition 11. Let (M'", g, V) be a Newtonian system with potential en-ergy. The Lagrange function (or Lagrangian) L : TM' R is the differenceof kinetic and potential energy,

L = T-Voir.


Theorem 17 (d'Alembert-Lagrange). A curve 7 : (a, b) -+ Mm is a motionof the Newtonian system (Mm, g, V) with potential energy if and only if theEuler-Lagrange equations hold:

dt (ex (y(t))) = 8x (7(t))

Proof. We prove this theorem again using local coordinates. The curve7(t) _ (x1(t), .. . , xm (t)) is a motion of the Newtonian system with potentialenergy if and only if it solves the system of differential equations

»i mik = - I jiìj - E Va9ak

ij=1 a=1

For brevity, we denoted by V. the partial derivative of the potential function,4G, := 8V/8x°. The Lagrange function is

L('y) =2

gijii2j - V(1'),i,7=1

and this leads to the differencemd 8L 8L d9ia 1 99.

mQ a a a

dt (8zi ax-*l 2 axi) x + 9iay + V; .a,;3=1 a=1

Multiplying the Euler-Lagrange equations by gik and summing over theindex i, this system of equations turns out to be equivalent to

Mm m

0 = xk + V 'k + M a9ia 109.eE i9 E ` 8x3 2 axi 9i-l i=1 13a. =

The claim now immediately follows from the formula for the Christoffelsymbols r .. o

13

Thus the equations of motion of Newtonian mechanics are, in the case of apotential force, equivalent to the Euler-Lagrange equations. For the latter,it does not matter that the Lagrange function arises as the difference of akinetic and a potential energy. Hence we define:

Definition 12. An autonomous Lagrangian system is a pair (Mm, L) con-sisting of a manifold Al" and a smooth function L : TMm - R. A La-grangian motion is a curve -y : (a, b) -- M" which solves the system ofEuler-Lagrange equations

Wt(er ('i (t))) = dL ('Y(t))


Example 11. Let (M'", g) be a pseudo-Riemannian manifold and A a 1-form on it. The Lagrange function

L(v) =2

g(v, v) - A(v)

generalizes, in a sense to be discussed in Chapter 9, the motion of a chargedparticle of mass m under the influence of the Lorentz force of the electro-magnetic field generated by A.

Motions in a Lagrangian system can be understood from the point of viewthat they are critical points of the action integral L. This measures for acurve -y : [a. b] -p Mm the mean value of the Lagrange function on thiscurve,

b

L(y) :=Ja

Theorem 18 (Principle of Least Action). A curve -y : [a. b] Mm is amotion of the Lagrangian system (Mm, L) if and only if the variation of theaction integral vanishes for every variation y1, of the curve with fixed initailand end points, y1,(a) = y(a), -y. (b) = y(b):

d

dµ

(jb)LO = 0.

Proof. We compute the derivative of the action integral with respect tothe parameter p in coordinates -t,, (t) = (x1(µ, t), ... , x'"(µ, t)) by partialintegration:

d (,C(-t,.)) 1,0=Jab [aL((t)) - d (a (7(t))) (0, t)dt.

The functions 8x'(0, t)/8µ are arbitrary functions vanishing at the endpoints of the interval [a, b], and hence the Euler-Lagrange equations areequivalent to the condition

(L(yµ)) Iµ=o = 0.dµ

For general Lagrangian systems, there exists a notion of energy which, onthe one hand, generalizes the energy of a Newtonian system with potentialenergy, and is, on the other hand, a preserved quantity. This Lagrangianenergy is obtained by first introducing the Legendre transformation. LetL : TM' R be a Lagrange function, and let v E .,Mm be a vector atthe point x E Mm. Now restrict L to the tangent space .,Aim and considerthe differential D(LIT=SIm)(v) at the point v E TTMm. This is a linear mapTTll1"' IR. hence a covector in T *M'.


Definition 13. The Legendre transformation C : TM' T'M' of anarbitrary Lagrangian system is the map

,C(v) := D(LIT=Mm)(v).

Example 12. If the Lagrange function L = Zg - V is the difference of akinetic and a potential energy, the following relation holds for v, w E TM':

G(v)(w) = 2 . D(g)(v)(w) = &,w).

Hence the Legendre transformation L : TM"' T* M' is simply the iden-tification of the tangent bundle with the cotangent bundle via the metric.

Definition 14. The energy of a Lagrangian system (MI, L) is the functionE : TM' -' R on the tangent bundle defined by

E(v) = L(v)(v) - L(v).

In the case of a Newtonian system with potential energy, we have

E(v) = L(v)(v) - L(v) 2g(v, v) + V(r(v)) .

This shows that the energy in the sense of Lagrangian mechanics coincideswith the Newtonian energy.

Example 13. We compute the Legendre transformation for the Lagrangefunction from Example 11. Let (x, y) E TTM" with local coordinatesxl... , xm and y', ... , y'. Then,

m m

...,y"') = 2 Egjjy'1!r - EAiy',1,3 i

and, as an element of T,M, its differential ism m

D(LITTMm)(yl,...,ym) = mEgify'dhe-EAidy'.

i,j i

We evaluate this map at (x, v), v = (v', ... , v'"):

G(y)(v) = D(LI T:MO1) (y)(v) = m . g(y, v) - A(v) .

By definition, this yields for the energy the value

E(v) = C(v)(v) - L(v) = 2g(v, v),

which has the remarkable property of not depending on A. A more carefulphysical analysis shows that the energy of a charged particle indeed onlydepends on the electric, not on the magnetic field. This is related to thefact that the magnetic field does no work on the particle (see §9.5).


Remark. Denoting the coordinates on TM' by {x', , X-, x', ...'e }and the coordinates on T` Mm by {qj, ... , qm. pi, ....p,"}. we see that theLegendre transformation f- is given by

aLqi = x' and pi =5p,and the expression for the energy E takes the following form:

_ "' OLE(x, x) - 8ii

xi - L(x, x) .

==Y

Theorem 19 (Conservation of Energy for Lagrangian Systems). The energyE(y' (t)) of each motion ti(t) of a Lagrangian system (Mm, L) is constant.

Proof. The energy of a curve is

E(7(t)) _8L dxi

dt - L(ti(t))t-1 f72i

and by differentiation we obtain

dE(Y(t))

"'

C

8L dx' dx- 8L dx' 1 ' &L dx'dt _ 8ii8xj dt dt + C7x'&i dt dtZ J - dxi dtij=1 i=Y

Using the Euler-Lagrange equation

i3Lm

c7L dx' "` OL d2xiOx-. =

ij=O±'Oxi dt + axiaxj dt2EY

i,7=Y

we immediately obtain the assertion. 0

Thus the energy is a first integral for any motion in a Lagrangian system. Asin §7.3. further first integrals can be derived from symmetries of the Lagrangefunction. To this end, consider a one-parameter group of diffeomorphisms,P6 : Al' M' of the configuration space as well as its generating vectorfield,

V(x) :=dd('ts(x))j8=0

on Al'. The differentials d(4 Q : TM' TM"' are diffeomorphisms of thephase space TM"' into itself.

Theorem 20 (Noether's Theorem). Let the Lagrange function L be invari-ant under the action of a one-parameter group of diffeomorphisms,L(d(4s)(v)) = L(v). Then the function fv : TM' --+ R defined by

fv(w) = limµ-.O µ

is a constant of motion for the Lagrangian system (Mm, L).


Proof. Let the one-parameter group of diffeomorphisms $t be determinedin coordinates by

4tt(xil ...,xm) = ('Ft(x1, ...,X"), .... Dm(x'....,xmThe invariance of the Lagrange function implies that

m aL a(De m aL O4 dx'ax= as I + E axj . ax=as dt

and the vector field V has the componentsm

aV = E s

x1a3

s=IUC7-

i

Thus we obtain

'

=

= 0,

fv(7(t)) = 1 8x as Is.o'and from the Euler-Lagrange equation we conclude that

d m IL 81' m aL 02V dxjdt = o. od ax= WS=0 I8=0 + ax' ax_as o

Example 14. To each transformation group'Ft : Mm - Mm preservingthe metric g and the potential function V there corresponds a first integral,

fv(w) = g(w,V),which is linear in every fiber of the tangent bundle Tlblt. We made useof this first integral in Theorem 37, Chapter 5, to integrate the geodesicflow on surfaces of revolution (Clairaut's theorem). Hence first integrals ofthe geodesic flow which are linear in the fibers arise from isometries of themetric.

Example 15. If 't preserves the pseudo-Riemannian metric g as well asthe 1-form A from Example 11, then the Lagrangian of a charged particlein an electromagnetic field is invariant, and hence Noether's theorem can beapplied to yield the invariant

fv(w) = m g(w, V) - A(V).

Formulation of Mechanics According to Hamilton

We will arrive at the formulation of mechanics according to Hamilton bystarting from a Lagrangian system (M, L) with bijective Legendre trans-formation C : TM' T'Mm. These Lagrangian systems are called hyper-regular. A regular Lagrangian system is one whose Legendre transformationis locally invertible. For simplicity, we confine ourselves here to the case


of a globally invertible Legendre transformation. For example, Newtoniansystems with potential energy or the system of a charged particle moving inan electromagnetic field (Example 11) are always hyper-regular.

Definition 15. Let (Mm, L) be a hyper-regular Lagrangian system withLegendre transformation C : TM' --+ T* M' and energy E : TM' --+ R.The function

H := EoL 1

defined on the cotangent bundle is called the Hamilton function (or Hamil-tonian) of the system.

Example 16. In the case of a Newtonian system (Mm, g, V) with potentialenergy, the Legendre transformation is determined by the formulas

(9Lqi = xi and pi= axi =

Inverting this transformation leads to

m

E gi.±a=1

m

x' = qi and i' = E gtnPa ,

U=1

and the formulas for the energy E, the Lagrange function L, and the Hamil-ton function H read as follows:

(1) E = 2 gijx'i +V(xl, ...,x'"),i j=1

m(2) L = 2 E gijz'xi - V(xl, ...,xm),

ij=1

(3) H = 2 E g''rpipj+V(gl,...,gm).ij=1

Through this change of phase space-i. e., replacing the tangent bundleTM"' by the cotangent bundle T'M'-we enter the realm of symplecticgeometry, since T*M'" always carries the symplectic form w = d9.

Theorem 21 (Hamilton's Theorem). Let (M'", L) be a hyper-regular La-grangian system. A curve -y : (a, b) -' AM"' is a Lagrangian motion if andonly if the curve G(ry) : (a, b) -. T' M"' is an integral curve of the symplecticgradient s-grad(H) of the Hamilton function.

Proof. The Legendre transformation C is defined byi OLqi = x, Pi = axi .

For its inverse map, we introduce the notation

x' = qi and i = I (ql, ... , qm, Pi, ... , pm).


The Hamilton function is thenm

H =a=1

and we compute its partial derivatives:M+

1: (P. -apt a=l

OH m aL aV _ OL

=_ OL

a aqt axt axt.a E51-

Thus we obtain a formula for the symplectic gradient:

A curve

t=l

G(?'(t)) = 1 21(t), ... , 2m(t), 'oil (fi(t)), - - , O L

is an integral curve of the vector field s-grad(h) if and only if

it = V and Wt(5pIL

(ti(t))) = axt ('Y(t))

The first equation is trivialy satisfied by setting it = fit, and this proves theassertion. 0

Exercises

1. In R4 with the symplectic structure w = dxl Adx3 +dx2 Adx4. we choosethe following four diffeomorphisms a, b, c, d:

a(xl,x2,x3,x4) = (21,x2+1,x3,x4),b(xl, x2, x3, x4) = (xl, x2, x3, x4 + 1),

C(xl, x2, x3, x4) = (xl + 1,x2, x3, x4) ,

d(xl, x2, x3, x4) = (x1, 22 + 24,x3 + 1,x4 ),

and denote by r the group of motions of R4 generated by them. Provethat w is a F-invariant 2-form. Hence w induces a symplectic form on themanifold M4 = R4/F. Prove that M4 is compact. Finally, denote by [F, F)the commutator group of F. Then F/[t, rJ is a free abelian group of rankthree (Thurston, 1971).

s-grad(H) = E(i+)

Exercises 265

2. Prove that the symplectic form w of a compact symplectic manifold M2mcan never be an exact differential form. Hence the second de Rham cohomol-ogy HDR(M2m) is non-trivial. In particular, for m 1, even-dimensionalspheres St' have no symplectic structure.

3. Consider M = R2\{0} with the symplectic form w = dx n dy and thevector field

x 8 y 8V _x2+y2 8x+x2+y2 8y

a) Let cp = arctan(y/x) be the polar angle defined in every sufficientlysmall neighborhood of (x, y) (0, 0). Compute grad(W) and

b) Conclude from this that V is no Hamiltonian vector field on all of M.

4 (Continuation of Example 3). Prove that the Liouville form on the two-dimensional coadjoint orbit through the element (a, Q) E g` (3 54 0) is givenby the expression

w = daAdf .

Hint: Show first that the fundamental vector fields corresponding to the Liealgebra elements (1, 0) and (0, 1) are

aa, and - Qom.

5. Let V be a vector field on a manifold Mm and 1 = V(x) the associateddifferential equation. A first integral of V is a smooth function h : Mm - Rwhich is constant along every solution of the differential equation.

a) Prove that h : Mm -p JR is a first integral of V if and only if dh(V) = 0.

b) Prove that the set Cv (M'n; R) of all first integrals of V is a subring ofC' (M'; R).

c) Show that h(x, y) = y2 - 2x2 + x9 is a first integral of the vector field

V = 2y 8/8x + (4x - 4x3) 8/ayin R2. Describe the geometric shape of the integral curves in the(x, y)-plane by means of h.

d) Find a vector field in the plane which has no non-trivial first integral.

6. Let f = a + b i : U -p C be a holomorphic function on an open subsetU of JR2 ^_' C. Consider the vector field V = grad (a).

a) Prove that b : U -. R is a first integral for V.


b) Describe the geometric shape of the integral curves of this vector fieldfor the functions f(z) = zk and f(z) = z + 1/z.

7. Consider on ]R3 a vector field B (a magnetic field) and its 2-form

B =as well as the symplectic form on the phase space R3 x JR3 with coordinates(9, ii) = (x, y, z, Vx, Vy, Vz),

wB = m(dxAdvx+dyAdvy+dzAdvz) - - B.C

As Hamilton function, we choose the kinetic energy

Ho = 2 (vZ+vy2+vZ).

a) Show that the defining equation for the Hamiltonian vector field associ-ated with Ho is equivalent to the Lorentz equation

dv _ e(*) mdt

c.vxB.

b) If B = dA is exact and A denotes the associated vector field, show thatthe map

f : 1R3 x JR3 -R 3 x 1R3, (4> ) ' (q, mii + A) (q, p)

is a canonical transformation, i.e., for the canonical symplectic form woand the Hamilton function HB,

wo = dxndpx+dyAdpy+dzndpzi HB = -lip- -All

the following equations hold:

f*wo = 'B, f*HB = Ho,and equation (*) does not change.

c) If B is constant, any particle moves on a helix. Hint: Interpret torsionand curvature as first integrals. If both these quantities are constant fora curve, then this curve has to be a helix (Exercise 6 in Chapter 5).

8 (Plane Toda Lattice). Consider in JR2 the second order differential equa-tions

i = -ex-y, y = ex-y

which in phase space (x, y, i, y) E JR4 are equivalent to t; = X with

X(x,y,i,y) = aa + ya - ex-y a + ex-y

TX ay ai ayThe energy E = (i2 + y2)/2 + ex-y is a first integral.

Exercises 267

a) Show that this system has further first integrals, for example P = i + yor K = (i - 2y)(y - 2i)/9 - e---y. These quantities are related byE + K = 5P2/18.

b) The setM2(E, P) (x, y, i, y) E 11 P4 : (i2 + y2)/2 + eZ-v = E, ±+y=P}

is not empty if and only if 4E - P2 > 0. In this case M2(E, P) is asmooth two-dimensional submanifold of R4 without boundary . Thisleads to a decomposition of ]R4 into a family of submanifolds, and everyintegral curve of X lies completely in one of them.

c) Show that each of the submanifolds M2(E, P) lies in an affine subspaceof dimension three and is diffeomorphic to R2.

d) If {(t) = (x(t), y(t), i(t), y(t)) is an integral curve of X in M2(E, P),then

i+y = P and x yf 4E-P -4ez-V = tShow that (A > 0)

dz _ 1 vA- A -BeeA -Bez

=v/-A (

In I + A -Bee '

and use this formula to integrate the equations for the integral curvesin M2(E, P) completely.

9 (Euler Equations). Let I : ]R3 - R3 be a symmetric positive definiteoperator. Consider the differential equation I(i) = 1(w) x w, where xdenotes the vector product.

a) Prove that this differential equation has two first integrals, the energy2E = (I(w),w) and the momentum M2 = (I(m), I(w)).

b) Conclude from this that the integral curves of the differential equationare intersections of two ellipsoids with center 0 E IR3 (I 0 IdR3). Inparticular, all integral curves are closed. Let Il be the smallest and 13the largest eigenvalue of I. Then there exists an integral curve only for2EIl < M2 < 2E13. .

10 (Motion in a Central Force Field). Let a potential force F act on a pointof mass one in R3,

FdU

drr) r -grad(U(r))with a function U(r) depending on the radius r = IIxil only. The energyE = 11±112/2 + U(r) is a first integral.


a) Show that M = x x . is a further first integral (M is the angularmomentum), and conclude from this that every trajectory of the pointlies in a plane in R3.

b) Determine for which values of the parameter M E JR3 the level surface

A3(M) = {(x,x)ER6:xx:i=M}is a three-dimensional submanifold of phase space R6.

c) Let r(t) be the distance to the origin of a motion whose an;;ular mo-mentum is M. Then

d2r dU 1IM112

dt2_ _dr

+ _73-i.e., r(t) describes the motion of a point in R1 under the effective forceF2 = -grad(V) with effective potential V(r) = U(r) + IIMII2/2r2. Theenergy of this motion is

E* = 2 + U(r) + I I2r2Iz

Prove that this energy E* coincides with the energy E (for fixed angularmomentum M).

Hence, if x(t) E R3 is the trajectory of a point moving under a force F andr(t) is its distance to 0 E K3, then

V2E -IIMII2 - 2U(r) = r', i.e. J

r -dt = t

11 (Classical Moment Map). Consider K3 with the defining representationof SO(3, K).

a) Prove that this representation is equivalent to the adjoint representationof SO(3, K) on its Lie algebra so(3, K), if R3 and so(3, K) are identifiedvia the map

vl 0 -V3 v::R3 so(3, f8), v = v2 u.- v = v3 0 -V1

V3 -V2 vi 0

Moreover, the adjoint and the coadjoint representation of SO(3, K) arealso equivalent.

b) Prove that this identification satisfies the equation

v(w) = v x w, [v, w] = [v, w], (v, w) Ztr (vv-) .

Exercises 269

c) By a), the moment map' : T'R3 - so(3,R)* can be interpreted as amap from T*R3 to R3. Show that it can then be written in the form

W(9,P) = 9XP,and hence it coincides with the classical angular momentum.

Chapter 8

Elements of StatisticalMechanics andThermodynamics

8.1. Statistical States of a Hamiltonian System

The Hamiltonian formulation of mechanics starts from a configuration spaceX"' and makes use of the phase space T'X' with its canonical symplecticstructure. A state of the mechanical system under consideration is a point inthe phase space T`X'", and the motions of states are the integral curves ofthe symplectic gradient s-grad(H) of a Hamilton function H : T* X' --+ R.In this formulation of mechanics, the only essential data to be given are asymplectic manifold M2rn and a function H. The point of view of statisticalmechanics is based on the idea that, for instance because of the size of themechanical system or as a consequence of the imprecision of measurements,the state of the mechanical system cannot be determined precisely by fixing2m real parameters. Instead, to each open set U C M2"', we can onlyascribe the probability p(U) that the state belongs to the set U. This leadsto the concept of considering mechanical states no longer as points in thephase space M2, but as probability measures on M2'".

Definition 1. Let a Hamiltonian system (M2"', w, H) consisting of a sym-plectic manifold and a Hamilton function be given. A statistical state is aprobability measure p defined on the a-algebra of all Borel sets on M2'".

271

272 8. Elements of Statistical Mechanics and Thermodynamics

Example 1. For each point x E M2n', consider the measure 62 concentratedat this point,

1 0 ifx¢U,a:(U) l 1 if a E U.

Therefore, classical mechanical states are particular statistical states.

Let -tt :M2" -+ M2in be the flow of the symplectic gradient s-grad(H).

The motion of the classical state x E MZ"' is the trajectory 0t(x). For theprobability measure b+,(s) corresponding to the point it(x), we have

60&)(U) = 110,, x

EE

xt

t (U1)

(U) 1 = bx(-tt I(u)),

and this formula leads to the following definition.

Definition 2. Let a Hamiltonian system (M2-, w, H) and a statistical statep be given. The motion of µ under the impact of the Hamiltonian systemis the curve At of measures

At (U) := µ(4 1(U))

Definition 3. An equilibrium state of a Hamiltonian system (M2m,W, H)is a state A which does not change in the course of the motion of s-grad(H),At = P.

Definition 4. A statistical state p of a Hamiltonian system (M2in,W, H)has a stationary terminal distribution if the equation

'U-(U) := limt MO)00

defines a Borel measure on M2"'.

Theorem 1. A stationary terminal distribution µ«, is always an equilibriumstate.

Consider the volume form

dM2m _(-1)m(m-1U2

Wm

m!

of the symplectic manifold and the Borel measure induced from it (see thefinal remark at the end of §3.5). If the measure p is absolutely continuouswith respect to the volume measure,

µ(U) := je(x).dM2m(x),

then a is called the density function of the state µ.

8.1. Statistical States of a Hamiltonian System 273

Theorem 2 (Liouville's Equation). Let p = be a state with densityfunction, and let at be the motion of this state in a Hamiltonian system.Then the measures pt = pt dM2" are also states with density function, and

ddtet = -{H,e}o$_t.

Proof. The flow ibt consists of symplectic transformations and preserves, inparticular, the symplectic volume form dM2in, 4 (dM2i`) = dM2m. Trans-forming the integral, we see that pt = P o &_t is the density function of thestate µt. This implies

dt of = dtP o D_t = do o dt -t = - o -t = -{H, p} o O_t.

0

Corollary 1. A statistical state µ with density function P is an equilibriumstate if and only if a is a first integral of the Hamiltonian H, (H, g} = 0.

If x E M2m is a classical state in Hamiltonian mechanics, the value H(x) ofthe Hamilton function H is the energy E of the state,

E(x) = H(x) = JM2rn

The definition of the energy of a statistical state generalizes this relation:

Definition 5. The energy of a statistical state p is the integral

E(µ) =JMom

H(x) dp(x),

if this integral exists.

Theorem 3 (Conservation of Energy in Statistical Mechanics). If u t is themotion of a statistical state in a Hamiltonian system, the energy E(pt) isconstant.

Proof. After transformation of/ the integral, we immediately obtain

E(pt) = l H o I 1(x) dtt(x),J M2m

and differentiating this with respect to the parameter t yields the formula

d E(µt) = fM2m {H, H}(x) du(x) = 0. 0

Now we want to study how the probability 1At(N2m) that the state µ =B dM2ri is in the compact subset N2ni C M21 at time t changes in time.The following lemma serves as a preparation.


Lemma 1. Let (M2i', w) be a symplectic manifold. Then, for any twofunctions f, g : lbl2in --+ R, the following formula holds:

df AdgAw,,,1 = 1 If,M

Proof. To prove this formula, we choose on 1112»' local syinplectic coordi-nates (ql, ..,P.),

w = dp0 A dqq,r= I

Setting Ai := dpi Adgi, we see that the exterior product satisfies the relations

AiAAj = AjAAi and AiAAi = 0.Since the forms Ai commute with one another, we can use the binomialformula to compute the exterior power wk,

m > 1

wn, _ (A1)' m.Aa' A A A"

Because Ai A Ai = 0, only summands with rri < 1 occur in this sum, andhence it reduces to a single term:

writ = in!.AlA...AAna = m!.(dpiAdgi)A...A(dpmAdq»,)We compute WI-1 in a similar way, and obtain

",

(m-1)! AlA...A,;...A

Thus, the following exterior products vanish:

(1) dpi Adpj Aw"'-1 = 0 = deli Adgj Awl"-

(2) dpi A dqj A w' = 0 if i # j

and we obtainfit

df A dg Awrr-1 = (.!m- dg dp, A dqi +2f ag

dq, Adpi) w"'api Oqi OCli dpi

rnOf yg Of 09('m - 1 ) !

p i T q i - di C d R

A I A ... A A,,,

arr.-1 . rmi {f,g} w

The relations m. d(f dg A w'"-1) = in df A dg A w"'- I = {f, gj w"' togetherwith Stokes' theorem imply, after integration.


Theorem 4. Let (A12mw) be a compact symplectic manifold with boundary.Then

{f,g} w"rJ12in

In particular, the integral

f.dg nwm-1i)At 2° .

{f,9} w"` = 0Jr2"

vanishes for any compact symplectic manifold without boundary.

Corollary 2. If the Poisson bracket if, g} of two functions defined on acompact symplectic manifold 1112" without boundary does not change sign,then it vanishes identically, If, g} = 0.

We will apply these formulas to a statistical state with density function e.

Definition 6. The (2m. - 1)-form

At) (m-1)!A'dHAw"'-1

is called the probability current of the statistical state it = e dM2"' in thesymplectic manifold (Al2m,w) with Hamilton function H.

Theorem 5. Let lit be the motion of the statistical state p = e dA12"' inthe Hamiltonian system (1112"', W, H), and let N2", C AI2si be a compactsubmanifold. Then

Wt(µr(N2-)) (r=o = JNFIJ

Proof. We compute the derivative with respect to time at t = 0 and useLionville's equation as well as the preceding integral formulas:

d N2", _ J dM2", J {He} dM2"ro - dt r=eN2m N2s'

(m - 1)! Je dH n w"'-i = f j(Ic) .

UN2"4 aN2"

The probability current j(µ) is a hypersurface measure on all (2m - 1)-dimensional submanifolds of the symplectic manifold M21. It expressesthe infinitesimal change of the probability for the state it to be in the setN2i' C A12"' as an integral over the boundary of N2'".

Example 2. If the subinanifold N2i" C M2" is described by inequalities ofthe Hamilton function,

N2n = {x E M2" : C1 < H(x) < C21,


then the differential dH vanishes on the tangent bundle T(8N2m) of theboundary, and so does the form j(p). We hence obtain, for the motion Atof every statistical state p = p dM2,,

dtpt({x E M2in : C1 < H(x) < C2}) = 0.

Thus the probability for a state p to be in N2in at time t is constant.

Example 3. In case m = 1, the formulas for the probability current of atwo-dimensional Hamiltonian system (M2, w, H) simplify:

j(p) = e dH and dpt(N2) = p dH.t JaN2

Now we turn to the notion of information entropy for a statistical state.As a motivation, we first recall the notion of information of an event in-troduced by C. E. Shannon (1948). The heuristics is the following: If, in aseries of experiments, an event occurs with probability close to one, then the"information" contents of this event is small. Conversely, if the probabilityof this event is close to zero, the occurrence of this event contains a largeamount of "information". Modeling the probability computation by a triple(i, 21, p) consisting of a set i, a or-algebra 21 of subsets of S2, and a measurep defined on 21 such that A(fl) = 1, we arrive at the following definition forthe Shannon information.

Definition 7. Let (S2, 21,,u) be a probability space. The amount of infor-mation contained in an event A E 21 is

1(A) := - log(p(A)).

For a finite set S2, one forms the mean of these information amounts andgets in this way the so-called information entropy.

Definition 8. Let (12, p) be a finite probability space. The informationentropy of this space is

S(Q, p) f 1(w) . dp(w) = - E pi log(pi) ,t i=1

where the set 11 consists of the elements {w1, ...,wn}, and pi := p({wi}) isthe probability for the event wi.

The information entropy is a measure for the uncertainty of the probabilityspace. In fact, if St = {wl, ... , w } consists of n points and we denote by p'the measure corresponding to the equidistribution, p'(wi) := 1/n, then thefollowing holds.


Theorem 6. The information entropy of a finite probability space (S),µ)does not exceed the information entropy of the equidistribution:

S(Q, Ft) <_ S(n, µ')

Equality occurs if and only if u = µ' is the equidistribution.

Proof. The proof is based on the fact that the function f (x) := x log(x)is convex on the half-line (0, oo), since

f'(x) = 1 + log(x) and f"(x) = x > 0.

The convexity of f then implies the inequalityn n

f pi <_ n

f (pi)

= n E pi log(pi )i=1 i=1 i=1

n

Using the relation pi = 1, one immediately deduces from it thati=1

S(1, lt) > pi log(pi) < -log (n) = S(S2,i=1

The formula S(12,,u) = - E pi log(pi) leads to the notion of informationentropy for a statistical state with density function.

Definition 9. Let (M2,, w, H) be a Hamiltonian system, and let a = edM2m be a statistical state with density function. We define its informationentropy as the integral

S(u) - JM2m P log(P) dM2m,

if this integral exists.

Remark. This notion of entropy is not without problems in statisticalphysics. The following result concerning the conservation of entropy claimsthat the information entropy does not change in the course of the motion ofa state. This is a good assumption for reversible processes (and processeswhich can be approximated by reversible ones), but cannot be applied tothe vast range of irreversible processes, in which the entropy increases. Forthis reason, we will apply the notion of information entropy S(µ) only toequilibrium states.

Theorem 7 (Conservation of Information Entropy). The information en-tropy S(pt) of a statistical state with density function is constant under themotion of a Hamiltonian system.


Proof. We differentiate the function S(Pt) with respect to the parameter tand use Liouville's equation:

dt S(At)JM2m

(d-tL-log(owL dM2-

_ ({HQ}l(Q)+{HQ}) dM2-.M2m

But since {H, log(e)} = {H, a}/e, we have

{H,e}+{H,g}.log(e),

and this yields

dt S(µt)It_o = JM2m{H, e log(e)} dM2m

(-1)m(m-1)/2

m JM

(-1)m(m-1)/2

( ) JMs"`dA/ d(e log(e)) A w"`-1

m-1Since M2m has no boundary, Stokes' theorem immediately implies the as-sertion.

Remark. In the case of a manifold M2"' with boundary we obtain

d (-1)m(m-1)/2

IOM2.dt S({pt)It(

e log(e) dH n

w, H) has several special equilibrium states.These are characterized as realizing the maximum of information entropy ina particular class of statistical states. We will discuss two of these states,the so-called Gibbs state (or the canonical ensemble) and the microcanoni-cal ensemble. We will, in general, make the following three assumptions forthe Hamiltonian system (M2m, w, H):

Assumption 1. The Hamilton function H is non-negative,

H(x) > 0 for all x E M2s` .

Assumption 2. For every positive number B > 0, the following integralexists:

Z(B) :=JMr e-N(x)/B. dM2m(x)

The function Z(B) is called the partition function of the Hamiltonian sys-tem.


Assumption 3. For every positive number B > 0, the following integralexists:

H(x) . e-H(=)lB . dM2m(W ) .bl2m

Remark. The parameter B will later be identified with absolute temper-ature (multiplied by the Boltzmann constant k). Its appearance here indi-cates the exceptional role that temperature plays among all thermodynamicparameters.

The function

E(B) := I H(z)e-H(:)/B , dA12-(z)Z(B) M2m

is called the inner energy of the Hamiltonian system (M2m, w, H); its innerentropy is

S(B) := log(Z(B)) + E(B)B

Finally, the free energy F(B) is defined by

F(B) := -B log(Z(B)) = E(B) - B - S(B).First we note that the inner energy E(B) is a non-decreasing function onthe interval (0, oo). Its derivative is

dE 1

TB_ [(IZ2(B) B

M2m

z)e-dM2m(x))e- dM2m(x)) (ML H2(

_ l2

)e dM2m(x))- (ML H(x

and the Cauchy-Schwarz inequality shows that this derivative is positive(H j4 const). Denote by En and Em., respectively, the bounds of therange of the inner energy E : (0, oo) - R. We calculate the derivative ofthe partition function in a similar way:

dZ

=1

dB B2 M2mJ BZ Z(B) E(B).

Hence the inner energy as well as the inner entropy can be expressed interms of the partition function. We summarize these formulas.

Theorem 8 (Simple Equation of State for a Hamiltonian System).

(1) E(B) = B2gR (log(Z(B)));

(2) S(B) = W(B log(Z(B)));


(3) in the sense of 1 forms on the one-dimensional manifold (0, oo),

dS = B dE.

Proof. The third equation is a straightforward consequence of the first two.a

The Gibbs state (canonical ensemble) is distinguished by the property thatit realizes the maximum of the information entropy among all stag of fixedenergy.

Theorem 9 (Gibbs State, Canonical Ensemble). Let an energy value Eobetween the minimum and the maximum of the energy function E(B) forthe Hamiltonian system (M2m, w, H) be given. Among all statistical statesp = P dM2' with density function of energy E(p) = Eo, there existsprecisely one state, PGibbs = PGibbs dM2m, of maximal information entropyS(pcibb.). The density function of this state is

1 -H(x)/BoPGibbs(x) = Z(Bo)e ,

where the parameter Bo is determined as a solution of the equation E(Bo) _Eo. The value of the maximal entropy S(pCibbs) is

S(pGibbs) = S(Bo) = log(Z(Bo)) + E(BO) o)

Proof. Choose Bo as a solution of the equation E(Bo) = E0, and )et

pGibbs = 1 e-H(x)/Bo dM2mZ(Bo)

be the corresponding Gibbs state. Its inner energy is

E(pcibb.) = Z(B0) JM2mH(x)e-H(-)/Bo . dM2m = E(Bo) = Eo,

and hence UGibbe is a statistical state of energy Eo. For every oti er statep = P dM2m with energy

E(IL) = f H(x)P(x) dM2'n = E0,M2m

we consider the function

f (t) = S((i - t)pGibbs + tp) ,

the information entropy of the statistical state (1 - t)pGibbs + tp. Differen-tiating f (t) with respect to the parameter t, we obtain

d2f(t) - _JU2-

(e - PGib.)2 dM2' < 0 .dt2 (1 - t)PGib, + tP


Moreover,

dfe - log(eGibbs) - dM2m - S(PGibbs),dt

= fy2"

and calculating the integral using E(p) = Eo leads to

JM2"'e log(gGibbB) - dM2"' = Jim g( go - log(Z(Bo))) dM

_ -log(Z(Bo)) - BoEo = -S(Bo)

We conclude that the derivative of f vanishes at t = 0. Hence the derivativeis non-positive, and f (t) is decreasing in [0, 1]. Since f (0) > f (1), we obtain

S(P) < S(lpGibbs) -

If S(µ) = S(PGibb.), the first derivative of f (t) vanishes identically. Hencethe second derivative of this function is also zero, and from the formulastated before we conclude that e = eGibbs The assertions of the theoremare proved. 0Example 4 (Maxwell and Boltzmann Distribution). In classical statistics,the Hamilton function of a particle with potential energy depending only onthe coordinates q can be written as

2 2 2

H(q, p) = 2m + U(q) =P1

+2m+ 3 + U(q).Thus, the partition function is equal to

rZ(B) = I exp [_i?

] dpiAdp2ndp3 f exp IqJ1

L

which, using the integral fR a-QS2dx = r/a, can also be written as

Z(B) = (27rmB)3/2 - ZZ(B) ;

Zq(B) denotes the value of the integral in which the unknown potentialenergy occurs. In the same way, the Gibbs state factors into the probabilitymeasure Pp of momentum values and that of coordinate values Pq,

PGibbs = lip 'A q = (27rmB)3/2e-p2/2mBdP Zqe-U/Bdq

Going from the momenta to the velocities, we arrive at the so-called Maxwelldistribution,

M )3/2exp [m(vl +v3)] dv1dv2dv3,Pp

2 2B

which again is the product of probability distributions for every single ve-locity component. The Boltzmann ensemble is defined to be the probability

282 8. Elements of Statistical Mechanics and Thermoc.ynalnics

measure p. and is proportional to the particle counting measure v. Forexample, for the potential U(x, y, z) = ingz of a. homogeneous gravitationalfield which is parallel to the z-axis, this yields the barometric formula

V = 1/0 t,-inyz/'idx. d y dz.

The second natural equilibrium distribution of a Hamiltonian systc n startsfrom a subset A C A12ni of the symplectic manifold which is invariant underthe flow 4Dt of the Hamilton function H and has positive finite volme,

4bt(A) C A, 0 < vol(A) < oc.

Consider the density function

j 1/vol(A) if x E A,`°A (x) = l 0 if .r E A,

as well as the statistical state ILA = eA 012m. This is an equilibrium state,since A is invariant under the flow 4bt. Its information entropy is

S(,A) = - f ll,PA(x) - log(PAW) d12i'(x) = log(vol(A)).

Theorem 10 (Microcanonical Ensemble). Let A C M2"' be an invariantsubset for the flow of the Hamiltonian system (A12", w, H) with positivefinite measure. If it = p dM2"' is a statistical state whose density functionhas support in A, supp(p) C A. then its information entropy is alwaysbounded from above by the information entropy of p.A,

S(Ez) < S(tA) .

Equality only occurs for it = AA.

Hence the microcanonical ensemble IAA realizes the maximum of the infor-mation entropy among all statistical states concentrated in the set A.

Proof. The proof proceeds in complete analogy to that of the precedingtheorem. Again, we consider the function f(t) = S((1 - t)IA 4 tit) andcalculate its second derivative:

d2f= fA

(e - UA)2 . dA12,n < 0.dt2 -(1-t)eA+p

The derivative of f at t = 0 vanishes:

df I = - f (e- -log(eA) f 0,dt t_tt A A

since It is a probability measure with support in A. 0

8.2. Thermodynamical Systems in Equilibrium 283

8.2. Thermodynamical Systems in Equilibrium

First we generalize the simple equations of state of a Hamiltonian systemby keeping a fixed symplectic manifold (M2",, w) but considering a family ofHamilton functions H(x, y) depending on an additional parameter y E N'.Let the parameter space also be a smooth manifold, and suppose that theHamilton function H(x, yo) satisfies the three assumptions of the precedingsection for any fixed value of the parameter yo E NT. Then the partitionfunction Z(B, y), the inner energy E(B, y), the inner entropy S(B, y), andthe free energy F(B, y) become functions on the manifold (0, oo) x Nk,

Z,E,S,F: (0,oo) x Nk -4 R,

and, of course, the equation of state B dBS = dBE holds for each fixedvalue of the parameter yo E The general equation of state of a Hainil-tonian system depending on parameters is a relation between the completedifferentials considered as 1-forms on (0, oo) x N".

Theorem 11 (General Equation of State for Parameter-Dependent Hamil-tonian Systems). In the sense of an equality for I-forms defined on themanifold (0, oo) x N", one has

dE =B

Proof. By the definition of inner energy, we have

B dS + S dB = log(Z) dB + B d(log(Z)) + dE,

and again inserting S, yields

dE.

Remark. If we allow the until now independent variable B to be a functionB : N" -> (0, oo) on the parameter space N", we may consider all thefunctions Z, E, S and F as functions on N' by means of the map

N' (0, oo) x N", y -' (B(y), y).

The exterior derivative commutes with the pullback of forms and thus im-plies on the manifold N" the equation

dE = B dS + E dB - B d(log(Z)).


The macroscopic description of thermodynamic systems starts from a fewassumptions. First, all thermodynamic systems should be in mechanicalequilibrium. Furthermore, it is supposed that no particle exchange takesplace during the change of states. Lastly, the macroscopic state of the ther-modynamic system is to be completely described by a parameter y E Nk,which is an element of a finite-dimensional manifold. If this holds, thefollowing phenomenologically observed laws are historically known as thefundamental theorems of thermodynamics. In a modern account of theoret-ical thermodynamics, they are taken as axioms:

Zeroth Fundamental Theorem of Thermodynamics. There exists apositive function T : N'' - IR, called the temperature.

First Fundamental Theorem of Thermodynamics. There exist two(not necessarily exact) 1-forms dA and dQ on the manifold Nr such thattheir sum

dA + dQ = dEis an exact form. The function E is called the energy of the thermodynamicsystem, dA is called the work form, and dQ is called the heat form.

Second Fundamental Theorem of Thermodynamics. The 1 -form

dQT = dS

is an exact form, and the resulting function S is called the inner entropy ofthe thermodynamic system.

At this level of abstraction, a thermodynamic system in its macroscopicdescription is a 4-tuple (NT, T, dA, dQ) consisting of a parameter manifoldN'', a function T : N' -+ R, as well as two 1-forms dA and dQ such thatthe 1-forms

dA + dQ and dQT

are exact. Of course, the following relation holds:

dE = dA + T dS ,

and this is Gibbs' fundamental equation of thermodynamics.

Now, with each parameter-dependent Hamiltonian system a thermodynamicsystem can be associated in a canonical way. In fact, let a symplectic mani-fold (M2n', w) and an additional manifold N'' be given. Fix two non-negative


functions,

B : N' (0, oo) and H : M2m x N' --10, oo) ,

and consider H as a family of Hamilton functions.on the symplectic manifold.Using the Gibbs state,

= e-H(z.b)lB(b) . dM21n(x)pGibt)s

we define the function T : N' --+ R1 and the 1-forms dA and dQ on N' asfollows:

(1) The temperature of the thermodynamic system is the function Bdivided by the Boltzmann constant k (= 1.380.10-23W/K):

T(y) := B(y)k

(2) The energy of the thermodynamic system is the inner energy of theib)Gibbs state Gbbs'uG

E(y) = E(B(y)) = H(x, y)e-H(x,b)/B(b) . dM2"'(x) .Z(B(y))

M2'"

(3) The entropy of the thermodynamic system is the inner entropy ofthe Gibbs stateuB(b)Gibbs*

S(y) = k S(iz Y) = k - log(Z(B(y))) + k E(B(y))B(y)

(4) The heat form dQ is the product of temperature with the differen-tial of entropy,

dQ :=(5) The work form dA is

dA = EdB - B d(log(Z)).

The First Fundamental Theorem of Thermodynamics,

dA + dQ = dE,

is, in the model just described, an automatic consequence of the generalequation of state for parameter-dependent Hamiltonian systems. Hence, bymeans of the Gibbs states, a thermodynamic system is associated with everysuch Hamiltonian system. Let V be a vector field defined on the manifoldN. Then

dA(V) = BZ


and, since

V(Z) = J V(-H/B)e-H/BdM2m = - JV(H)e/I5dM2m+Y-E.Z,

,%12m M2m

we obtain an explicit formula for the work form,

dA(V) =Z

Jwzm V(H) - e-B dM2m .

The pressure pi, of the thermodynamic system generated by the vector fieldV on the parameter space Nr is the function

PV (Y) - I V(H(x, y)) e H(=.v)/B(y) . dM2-(x) .Z(B(y)) M2"

Thus the relation dA(V) = - pv holds by definition, and the work formcan be represented with respect to a frame V1, ... , Vr of vector fields of themanifold Nr in the form

rdA = - PV. Qi,

i=1

where a,, ... , ar is the dual frame of 1-forms on Nr.

Example 5. The pressure p as used in thermodynamics is obtained if theparameter manifold N' is one-dimensional and describes only the volumeV, and V = O/8V is chosen as the vector field. It is known that then theonly contribution to the form dA comes from the mechanical work of volumechange,

dA = -p - dV.Considering in the continuum limit for a large number of particles their num-ber as a continuous variable, we see that the amounts of matter nl, ..., nrof a system consisting of r different chemical components can be chosen asparameters for the manifold Nr. In this case, the chemical work is tradi-tionally expressed by the so-called chemical potentials µl, ... , µr (which thereader should not confuse with statistical states):

rdA = pi dni.

i=1

The chemical potential µi of the i-th matter component is the negativeof the pressure associated with the vector field O/Oni. Similar formulasexpress the electromagnetic work in the presence of magnetic or electricdipol-momentum densities.

Consider, in particular, the case that there exists a vector field V on theparameter space Nr such that

V(T) - 1 and V(H) - 0.


All the thermodynamic quantities are then computable starting from thepartition function

Z e-H/kTdM2mM2m

or the free energy derived from it,

F = -k T log(Z) .Theorem 12. Suppose that there exists a vector field V on the parameterspace Nr such that V(T) = 1 and V(H) = 0. Then:

(1) S = -V(F) = -dF(V) = -V i dF.(2) E = F-T-V(F) =(3) dQ = - T Lv(dF).(4) dA =

(5) The pressure generated by an arbitrary vector field W is

pw = - dF(W) + dF(V) dT(W) .In particular, the vector field V generates no pressure, pv - 0.

Proof. Differentiating the partition function Z with respect to the vectorfield V, we obtain

V(Z) T2 J He-H/AT . dM2m = E - Zk M2m

k.T2

This implies the following formula for the derivative of the free energy,

V(F) = -k log(Z) - k Tk

T2 = -k (log(Z) +k

T) = - S,

and the first of the assertions is proved. The relation

S = -V(F) = -(V J dF)

impliesdS = -d(V J dF) = -,Cv(dF).

The expression for the heat form is calculated from its definition,

dQ = - T Lv(dF) .The energy E is determined by the free energy F as well as the entropy,

E = F-T(VJdF).Lastly, we calculate the work form:

dA = dE - T dS = d(F - T(V J dF)) + TLv(dF)= dF - (V J dF)dT.


Example 6 (Ideal Gas). We view an ideal gas as n points of equal massM in a volume region V C 1R3 which do not interact with one another.The configuration space of the mechanical system consists of the productof n copies of the set V factored by the action of the permutation groupconsisting of n! elements. If we want to avoid this group identification, wecan simply consider the phase space M := T'V x ... x T'V equipped witha measure which already takes into account this identification:

3n

dM :=i=1

The Hamilton function of the mechanical system is

i2111

X(91,Q2,43).

Here X denotes the characteristic function of the set V C R3. Calculatingthe partition function of the thermodynamic system depending only on thetwo parameters T and V leads to the formula

Z(T V) = = 1 J el2MkTdp/I13n

je_hh1TdcAdpm ` oo

n3n/2;j (27rkTM) .

We apply Stirling's formula log(n!) n (log(n) - 1) and calculate all thethermodynamic quantities for the ideal gas using this approximation. Thefree energy is

F = - k T log(Z) nkT (log(V/n) + 1 + 3 log(2akTM)) .

We obtain the entropy and the energy from the free energy according toTheorem 12, since the Hamilton function does not depend on the tempera-ture:

OF 3S = - = nk (log(V/n) +2

log(T) + coast) ,

OF 3nE =O

The pressure p of the ideal gas corresponds to that of the vector field de-scribed by the volume, and is determined by

OF nP = - 8V =

Here n/V is the particle density of the ideal gas.


Example 7 (Solid Body). A solid body consists of n points oscillatingaround a point. If we take this point to be the origin, the Hamilton functionis

n2 2 2H (p3i-2 +i-1 + 3i + K2 3i-2 + q3i-1 + q3i- ` \ 2Mi 2

i=1

Introducing the basic frequencies

v Ki

2,r Mi'we obtain as the value of the partition function

3n 1Z (k T)v3 .... yn

The free energy is

F = - k T log(Z) kT (n log(n/e) + 3n log(kT) + const) .

Starting from this, one can calculate the other thermodynamic quantities,for example the energy:

T2 49(7-F,) =E = F-T

5T_ -

5TThis formula says that every degree of freedom of an atom carries the meaninner energy k T, a result known as the Dulong-Petit rule. However, attemperatures close to absolute zero, this rule no longer holds, since themodel for a solid body as being a combination of harmonic oscillators is notadequate anymore.

The inner and the free energy are examples of so-called thermodynamicpotentials. Classical thermodynamics usually adopts the point of view thatthe choice of a particular potential already determines the quantities bywhich the system is to be described. In this sense, the inner energy E isunderstood to be a function depending upon entropy and volume, the freeenergy F is a function of temperature and volume, etc. Consider a Hamiltonfunction not depending on temperature. As in the two preceding examples,the free and the inner energy are related to the vector field V dual to thetemperature via the relation (Theorem 12)

E = O

Hence the inner energy E appears to be the Legendre transformation withrespect to temperature of the free energy chosen as Lagrange function, L :_-F (see §7.5). By means of Legendre transformations with respect to otherparameters, one obtains the other thermodynamic potentials, enthalpy andfree enthalpy, which we will not discuss in detail here. In the case that only


the mechanical work of the volume change contributes to the work form, theFirst Fundamental Theorem reads as follows:

dE =

T = OEaS and

p=

OE-VV

If the energy is even twice continuously differentiable, the interchangeabilityof second partial derivatives is equivalent to a so-called thermodynamic orMaxwell relation

02E OT Op

avas = aV = OSBeginning with the free energy,

dF =we similarly obtain the equations

S , p= -aV,as well as the thermodynamic relation

aS apaV aT

Eventually, we want to discuss the Carnot cycle briefly, which will lead usto the notion of universal thermal efficiency. Historically this was essentialfor the understanding of the First and Second Fundamental Theorems ofthermodynamics.

We use an arbitrary homogeneous substance and suppose that its state isdetermined exclusively by the two mechanical variables, the pressure p andthe volume V, from which the temperature T can be computed by meansof a general equation of state. A Carnot cycle is understood to be a closedpath ry in the (V, p)-plane consisting of two isotherms and two adiabatics.More precisely:

(1) From A to B, the substance is expanded up to the volume VBthrough contact with a heat reservoir of temperature Ti while keep-ing the temperature fixed ("isothermally"). During this ?rocess itabsorbs the quantity of heat Q1.

(2) The substance is taken out of the bath Tr at B and expanded tothe point C without absorbing or emitting heat ("adiabatically").The temperature drops to the value T2.

(3) The material is put into a heat reservoir of temperature T2 andcompressed isothermally from C to D, emitting the quantity ofheat Q2.


(4) The material is taken out of the second heat reservoir and adia-batically compressed from D to A to reach the temperature T1again.

It has to be stressed that we are supposing that the idealized processesdiscussed throughout this chapter are reversible. In the way explained here,the Carnot cycle describes a power engine (the temperature difference ofthe heat reservoirs is used to do work); reversing the process, it providesa thermal engine (work is done to further cool down the colder of the twoheat reservoirs). By assumption, work is done exclusively in its mechanicalform; the total work A is computed as

ipdV,A := jdA = -7

and, since -d(pdV) = dVAdp, this is equal to the area of the region boundedby the curve y in the (V,p)-diagram by Stokes' theorem. We integrate theFirst Fundamental Theorem

dE = dA + dQ

over the closed path -y. The line integral of the exact form dE along 7vanishes, since the inner energy again reaches the initial value at A:

0 = jdE _ 5dA+ ¢ dQ.ry 'T y

The thermal efficiency q is defined as the ratio of the work done and theadded heat, q := A/Q1. But the line integral along dQ is precisely Q2 - Qi;hence we can write the thermal efficiency also as

q=1-Q1


As the Carnot cycle was supposed to be reversible, the entropy does notchange, and the Second Fundamental Theorem implies the identity

Q 2 QiJry T T2 T1

The thermal efficiency only depends on the temperatures of both heat reser-voirs.

n =T1 - T2

T2

and not on the particular kind of substance or the particular constructionof the power engine.

Exercises

1 (Gay-Lussac Experiment, 1807). Consider a fixed amount of an ideal gaswhich at time 1 has pressure pl and volume V1. The gas is heated whilekeeping the volume fixed until it reaches at time 2 the pressure p2. Thenthe gas is allowed to expand adiabatically up to the volume V3 ("overflow").At the point 3. the gas reaches again the initial pressure pl. Prove that theentropy change between the states 2 and 3 is equal to

AS = S3-S2 = n-k-logV2

independently of the way in which the physical process is realized in practice.

2. Derive from the Maxwell distribution the probability distribution for theusing spherical coordinates.speed lit' I I = (vi + v2 + V32)112

3. Consider two containers filled with the same ideal gas having equal tem-perature To and particle number N, but different volumes V1 and V2. Thecontainers are then connected.

a) Calculate the equilibrium temperature of the connected containers fromthe condition that the entropy remains constant during the connectionprocess.

b) Explain why the work done is equal to the difference of the energiesbefore and after the connection, and calculate the latter.

Exercises 293

4. Determine the quantity of heat Q absorbed by an ideal gas in the courseof the Carnot cycle. Hint: One of the intermediate results is

Q = (T2-T1)(S2-S1)

5 (Van der W4aals Equation). The free energy of a real gas is heuristicallydefined as the sum of the free energy of the ideal gas F;d and an additionalterm depending on two constants a and b to be determined experimentally:

\F = F;d - nkT log V - nb n2a( V ) F2

a) Explain why this Ansatz tends for large volumes to the free energy ofan ideal gas, and why, on the other hand, it prevents the gas from beingcompressed unboundedly.

b) Derive the equation of state (the so-called van der Waals equation)

n2a(p +V

)

(V - nb) = nkT

as well as the formulas for entropy and energy. How can the result forthe energy be interpreted physically?

6. The Hamilton function of an ideal gas in the ultra-relativistiiyc limit is

n

(c ist the velocity of light). Calculate the free energy F, the entropy S, andthe energy E.

7 (Planck's Radiation Law). An ideal photon gas in thermal equilibriumemits black-body radiation, which is described by Planck's law. We describeeach photon by a quantum mechanical oscillator and assume as a knownfact that the energy levels of an oscillator with basic frequency w are

En = hw(n + 1/2). n E N .

For photons, it may be supposed that these are distributed within eachenergy level (up to irrelevant factors of proportionality) according to theGibbs distribution Qn = e-E^1kT

a) Calculate the mean energy,x x

On - En On

n=1 n=1

and conclude that the number n = [eF 1kT - 11-1 can be interpreted asthe mean occupation number.


b) The frequency w is related to the momentum of the photons by E =clIpIl = hw. Quantum statistics postulates that in the Maxwell distri-bution j 4p, the Gibbs factor e-`tIPII/kT is to be replaced by the mean

QA[ ofoccupation number n. Conclude that the probability measure poI IpI I, up to some factors, is

2Qaf

wuP - et-1k]

the spectral energy density c is defined as energy per volume unit, henceit is obtained after again multiplying by hw:

hw 3

e = efiW/kT _ 1

This is Planck's radiation law (Planck, 1900).c) Discuss the behavior of the curve c depending on the variable w and the

parameter T; describe, in particular, how the position of the maximumchanges with T.

d) Discuss the limiting cases hw << kT (Rayleigh-Jeans law) and hw >> kT(Wien's law).

Chapter 9

Elements ofElectrodynamics

9.1. The Maxwell Equations

The Maxwell equations describe the impact of an electromagnetic field on adistribution of electrical charges in space as well as the interaction betweenthe electric field E and the magnetic field B. These vector fields are time-dependent vector fields defined on a domain 11 C R3,

E: 1lxR---+R3, B: IlxR-.R3,and the electric charge is described by a time-dependent density function

e: QxR -*R.The electromagnetic field induces a current of the electric charges, and hencealso a time-dependent current density vector

J:S2xR-bR3.The operations of divergence and curl are supposed to be applied to the

spatial coordinates of the time-dependent vector fields.

First Group of the Maxwell Equations:

curl(E)

-- , div(B) = 0.

Second Group of the Maxwell Equations:

curl(B) = c- +4c J, div(E) = 47rg.

295

296 9. Elements of Electrodynamics

Here c denotes the constant velocity of light. Since div(curl(B)) = 0 (see§2.3, Theorem 5), the continuity equation

div(J) + = 0

is an immediate consequence of the second group of the Maxwell equations.

Now we want to write the Maxwell equations involving differential forms onR3. For this, we use the transition from a vector field V to the associated1-form wv discussed in Chapters 2 and 3. On a manifold Mm, wv is the1-form defined by the equation (see §§2.3 and 3.4)

*wv = V J dMm,

and the gradient and divergence satisfy the identities

Wgrad(f) = df, d(*wv) = div(V) - dAIm.

In JR3, there is an additional relation involving the curl of a vector field V(§3.11):

wcurl(V) = *dwv.Passing now from the time-dependent vector fields E and B to the corre-sponding time-dependent 1-forms WE and WE, we can restate the Maxwellequations in the following equivalent form:

First Group of the Maxwell Equations:1 (9

*dwE _. c 8t

(we). d(*wB) = 0.

Second Group of the Maxwell Equations:l e 47r

*dWB = -Cat

(WE) + - wJ, d(*wE) = 41rgo - dR3.

In these equations, the exterior derivative exclusively refers to the spatialcoordinates, and not to the time coordinate. By applying Poincare's lem-ma, we can represent the closed 2-form *wB locally as the differential of atime-dependent 1-form A,

*wB = dA.

Definition 1. The (locally) defined time-dependent 1-form A is called themagnetic potential.

The first Maxwell equation can be rewritten equivalently as

d(wE + c (A)) = 0.

9.1. The Maxwell Equations 297

Again, Poincare's lemma shows the local existence of a time-dependent real-valued function 0 with

WEc at

A -do.

Definition 2. The (locally) defined time-dependent function ¢ is called theelectric potential.

The essence of the electromagnetic field is not the same in classical elec-trodynamics and in quantum mechanics. In classical field theory, only themeasurable field strengths E and B are physically relevant; the potentialsA and ¢ are auxiliary mathematical functions which turn out to be useful.In quantum mechanics, however, the electric and the magnetic potential ac-quire a physical meaning on their own; we illustrate this briefly by discussingthe Aharonov-Bohm effect.

Remark (Aharonov-Bohm Effect). An infinitely extended coil S. throughwhich a current flows, induces a magnetic field which is almost homogeneousin its interior and vanishes outside of the coil. Consider a closed curve -yaround the coil and denote by 0 the surface bounded by ry. Imagine thatan electron is prevented from moving inside the coil by an infinitely highpotential well. Then the remaining region ci - S of the plane is, from thepoint of view of the electron, no longer simply connected. The vanishing ofthe magnetic field B in this region implies that the 1-form A of the magneticpotential is closed in 11 - S,

0 = *wB = dA.

but nevertheless the line integral is not path-independent, since, for example,the curve segments ryl and rye of ry cannot be deformed into one another by ahomotopy (see Theorem 9, §2.5). Instead, by Stokes' theorem, the integral

1A= JA = fdA = fis equal to the magnetic flux through the entire region Q. Quantum me-chanics describes the states of the electron by solutions of the Schrodingerequation called wave functions. Let V51 and v/2 be the wave functions of theelectron along -ti and rye, respectively, while the current is switched off (i.e..B = 0 in S). The covariance principle for the Schrodinger equation withrespect to gauge transformations implies that, if the current is switched on,the wave functions have to become

01 exp I AI , V2 = 02 exp [ , I AJL 7i rr


We put a screen at Si and form the linear superposition of the two wavefunctions:

B

+Gi + 1Pz = 'Pt exp [Z 2 ] + 02) eXp A]

As we vary the enclosed magnetic flux 4'O, the absolute value IV)*, + t21changes. This corresponds to a shift in the interference picture on the screen,and provides experimental evidence for the intrinsic meaning of the potentialA.

A theoretically rigorous explanation of this effect (A does not exist globallyon the region fZ!) can only be obtained within the theory of connections onprincipal S1-fiber bundles. The integral of A over a closed curve can thenbe defined correctly and measures the holonomy of the connection'.

If we put the potentials A and 0 at the beginning of our study, the electricand magnetic fields are determined by the equations

10WE c a (A) -do, wB = *dA,

and the first group of the Maxwell equations is automatically satisfied. Inthese formulas, we view the quantities as time-dependent functions or formsin 3-space, and the Hodge operator * and the exterior derivative d refer tothat space. We use the adjoint operator 5 of the exterior derivative d andnote that in R3 the following formulas hold:

8 = -*d* for 1-forms and 8 = *d * for 2-forms .

Then we can write the second group of the Maxwell equations as2

8dA1 8 (A) - 1 8 (do) + w.J,

1a

8A) O = 1aBc2 8t2 c cat c c tit (

The Laplacian A on functions will be used in this chapter analogously toChapter 2. Hence we have Ao = *d * do = -8d¢.

1As a physics reference we recommend: Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959),485-491, as well as the more detailed discussion of this effect in the textbook by F Schwabl,Quantum Mechanics, Springer, 2nd rev. ed. 1995.

9.2. The Static Electromagnetic Field 299

9.2. The Static Electromagnetic Field

If the electric and magnetic fields are time-independent, the Maxwell equa-tions simplify:

*dWE

*dwB

= 0, d(*WB) = 0,

4a- WJ,C

d(*WE) = 4ap d1R3 .

Hence the charge density P and the current density vector J do not dependon time, and J is a divergence-free vector field. This system of equationsdecomposes into a pair of partial differential equations for the electric fieldand another similar pair of equations for the magnetic field. We begin bysolving the equations for the electric field. Using the locally defined electricpotential WE = -do, this system reduces to the inhomogeneous Laplaceequation

-AO = 47re .

From now on we suppose that the support of the charge density is compact.Then it is to be expected that the electric field E(x) generated by this chargedistribution tends to zero at infinity. Under this condition, the electric fieldis uniquely determined.

Theorem 1. Let the support of the charge density p be compact. Then thereexists precisely one electric field E such that

div(E) = 4ae, curl(E) = 0and IjE(x)II - 0 for Jlxii oc. This electric field is given by E = -grad(4)with the electric potential

0(y) dy,OW11x - Y11

R3

which is also called the Newton potential for the density e.

Proof. Consider two electric fields El = -grad(01), E2 = -grad(O) gen-erated by one and the same charge density p. The difference u := 01 - 02 ofthe electric potentials is a harmonic function with a gradient vanishing atinfinity,

Au = 0, 1Igrad(u)(x)iI - 0 for IIxil -+ oo.

Since i (Ou/Ox') = 8(Du)/ax' = 0, the partial derivatives au/8x(1<i < 3) are harmonic functions on 1[t3 also vanishing at infinity. Liouville'stheorem for harmonic functions (see Theorem 42, §3.10) immediately impliesthat all partial derivatives vanish identically. Hence u is a constant function,and the electric fields El =- E2 coincide. There exists at most one electric


field with the required properties, and it remains to be proved that theelectric field E = -grad(O) with potential

OW = - P(y)dyf lix - yII

R3

is a solution for the problem. To see this, we have to show that O(x) is asmooth function in R3, that it solves the inhomogeneous Laplace equation-0O = 41rp, and that the gradient l grad(O)II -' 0 at infinity in R3. Weintroduce spherical coordinates around a fixed point x = (x1, x2, x3) E R3(0 < cp < 27r, -a/2 < ip < 7r/2):

y1 = x 1 + r cos cp cos ? / i , y2 = x2 + r sin c p cost , y3 = x3 + r sin Eli .

The volume form of euclidean space is described in these coordinates by thefollowing formulas:

dR3 = dy1 A dy2 A dy3 = r2 cos /i dr A dcp A d/i.

For every positive number e > 0, the function1

Ay) :=IIx -

yii3_e

is integrable in a neighborhood of its singular point. Integration over theball D3(x,1) with center x E R3 and radius 1 yields

r ffJ1 2,r K/2

J f (y) dy = r3-E cos i,i dr dcp dzb = 4a/e.

D3(x,1) 0 0 -7r/2

Because of the compactness of the support of the charge density g(y), thefunction c(x) and its first and second partial derivatives are smooth,

aox( ) - /'

P(y)x' y` dy = - /' P(y)

eax' - IIxyII3 J ayi (oxYII) dy

= f ay= (y)II x 1 yII

dy

R3

Differentiating once again leads to the formula

- AO(x) = lim lim f 0(P)(y) 'IIx

1

yIIdy.

MM-00 E-0D3(x.M)\D3(x,e)

Applying Green's second formula for sufficiently large radius M, we canrewrite the integral over the spherical shell D3(x, M)\D3(x, E):

f o(P)(y)FIX 1 yII

f P(y) ' bIIx 1 y1I

D3(x,M)\D3(x,e) D3(x,M)\D3(x,M)


- f [(y)(gy1 1 Y11, N(y)) -IIx 1 YII

\grady(y), N(y))] dy.

8D3(x,e)

The function 1/IIx - yUI is harmonic in the spherical shell: hence the firstintegral vanishes. Moreover, the normal vector field of the sphere 8D3(x, e)of radius a is given by the formula N(y) = (y - x)/E, which immediatelyimplies

1 1(grads

IIx - Y1I ,V(y))

E2

Inserting these expressions, we obtain

f OP(y)II 1 y1I dy = 2 f y(y)dy+ 2 J (grad e(y), y-r)dy .x- e

D3(x,A1)\D3(x.e) aD3(Xx) aD3(x,e)

The second integral on the right-hand side is bounded by

(1 /E2) - e - max{ Ilgrad oII } vol(8D3(x, E)) = 4;r - e max{ Ilgrad oII } ,

and hence it does not contribute in the limit as I 0. Together, this yields

-O0(x) = limo F J B(y)dy = 4irp(x),0D3(x.e)

where we used the mean value theorem of integral calculus in the last step.We estimate the length of the gradient of the electric potential using theformula for the partial derivatives 8o/0x` and for points lying far out. Todo so we choose a radius R > 0 such that the ball with center 0 E R3contains the support of the charge density. If now IIxII > R, then

23 x{ _ yi

Ilgrad6(x)II2 = f0(y)Ily-x113dyi=1 D3(0,R)

<3-max{p2(y):yER3}.4rr 1

3 dist2(x. D3(0; R))

We conclude that the length of the gradient IIgrad 4(x)II decreases like 1/IIxIIfor IIxII oo

Remark 1. Obviously, the preceding theorem and its proof remain validfor a charge density p which no longer has compact support, but, togetherwith its derivatives, vanishes sufficiently fast at infinity.

Example 1. Consider for a fixed point xo E R3 a charge density e which isconcentrated in a small ball around it,

ee(y) _ f e (vol(D3(xo,E)))-1 if y E D3(xo,E),0 otherwise.


We calculate the electric potential:

(x)Lot(Y) dy = -e

dy,IIx - Y11 vol(D3(xo, e) I IIx - Y11

R3 D3(xo,e)

and in the limit as e -+ 0 we obtain

lim , (x) - -e

E-o ` IIx - x011

In this way, the Coulomb potential arises naturally as the field generatedby an electric point charge. Let, similarly, n points xi, ... , x,, with chargese1, ... , e be given. Then they generate the potential

n

O(x) = - E ej

i=i IIx - xcll

Now we turn to the solution of the two partial differential equations forthe static magnetic field B(x) and consider again only the case that thedivergence-free current density vector has compact support (see the remarkabove).

Theorem 2. Let J be a divergence free vector field with compact support inR3. Then there exists precisely one magnetic field B such that

div(B) = 0, curl(B) = 4a Jc

and IIB(x)II - 0 for IIxii - oo. This magnetic field is given by B = curl(A)with the magnetic potential

lA(x)J

dy.c IIx -Y11

R3

Proof. Let B1 and B2 be two magnetic fields with the stated properties.Their difference V := BI - B2 is a divergence- and curl-free vector field,

div(V) = 0, curl(V) = 0,

whose length II V(x) 11 tends to zero for IIx11 -+ oo. By Poincare's lemma, sucha vector field can be represented as the gradient of a harmonic function f ,

V = grad(f) and Of = 0.The partial derivatives Of /8x' (1 < i < 3) are thus bounded harmonic func-tions on the entire space R3, and by Liouville's theorem they are constant.This immediately implies V = Bl - B2 = 0, and this observation shows thatthere can be at most one magnetic field with the required properties. By


the same arguments as in the proof of the preceding theorem, the magneticpotential

A(x)J

AY) dyc11X - Y11

R3

is smooth. It remains to be shown that it is divergence-free. The partialderivatives of the components A = (Al, A2, A3) of the magnetic potentialare

PON J`(y) (x' -1/') dy,c llx-yll3

,

R3

and hence

div(A) = Z' J (J(y),IIx - y113

)dy = I f (J(y), grad, (IIx1 yIl)

)dy.

R3 R3

Using the formula

divy(lix1 Y11 J(y)) = (J(y),g'ady(llx 1 y1I)) + I1x 1

YIIdiv(J)

and the fact that J was supposed to be divergence-free, we obtain

div(A) =

c

f divy(llx 1 YIIJ(v))dy = 0R3

by Green's formula, since the vector field J(y) has compact support. Thecurl of the vector field B = curl(A) can now easily be calculated (see Exercise6.b, Chapter 2):

curl(B) = curl(curl(A)) = grad(div(A)) - 0(A) = -0(A)

Here the Laplacian is to be applied component-wise to the vector field A,and the proof of the preceding theorem yields the result

curl(B) = -i(A) = 47r J.c

Example 2. Let the current density vector J be constant in a domainft C H3 and zero outside this domain. The magnetic potential and themagnetic field B = curl(A) are then determined by the formulas

A(x) c (f II X 'YII) J' B(x) c J " f II(x - 11 dy.n n

The magnetic field induced by an electric charge distribution with constantcurrent density vector is perpendicular to the direction of the current flow.This fact is called the Biot-Savart law.

304 9. Elements of Elect rodvnamics

Finally, we interpret both theorems as the Hodge decomposition of a 1-formon R3 (see Corollary 5, §3.11). Let wl be a 1-form on II23 with compactsupport or, more generally, with rapidly decreasing coefficients. ApplyingTheorem 1 to the function p = *d(*wl), we obtain a closed 1-form wE with

dwE = 0 and * d(*wE) _ *d(*w') .

The form wE also tends to zero at infinity, and hence, by Theorem 2, thedifference J = w' - WE can be represented as

*dw' = J = w' - wE

with a 1-form wa vanishing at infinity. This observation leads to

Theorem 3 (Helmholtz's Theorem, Hodge's Theorem). Every smooth 1-form w' in R3 with compact support, or, more generally, with rapidly de-creasing coefficients, can uniquely be represented as the sum of a closed anda coclosed form.

w' = WI +*dwg, L4 = 0.wF and 4 both tend to zero at infinity.

Remark 2. Comparing this result with Corollary 5 in §3.11, we stress that,in I3. the harmonic part does not occur in this decomposition. The reasonfor this is that there are no harmonic 1-forms in 1R3 having compact supportor tending to zero sufficiently fast at infinity. If w' were such a 1-form, then,by Poincare's lemma, we could represent w' as the derivative of a harmonicfunction, w' = df,zf, = 0. The partial derivatives Of 149x' would then bebounded harmonic functions, and thus constant. If then the length of w'tended to zero at infinity, then these partial derivatives would vanish andwe could conclude that wl = 0.

9.3. Electromagnetic Waves

An electromagnetic wave is an electromagnetic field in the vacuum (p = 0..1 = 0). The Maxwell equations in this case read as

div(B) = 0, div(E) = 0,

149E 149Ecurl(B) = -8t

, curl(E) c 8t ,

which implies that the components of the electric and magnetic fields haveto be solutions of the wave equation

AB = 1 02B AE _ 1 492E

c2 &2 ' cz 8t2 .

9.3. Electromagnetic Waves 305

If the wave is known at a certain time t = 0 and we want to study itspropagation in time, we have to solve the Cauchy initial value problem forthe wave equation:

102Du =

c2 at2 ,u(x, 0) = uo(x), - (x, 0) = ui (x) .

For smooth initial conditions uo and ul, this problem has precisely onesolution, which we are going to describe now. To do so we need the sphericalmean of a function V : 1R3 --i R. This is understood to be the functionIV : 1R3 x 1R+ -+ 1R which computes the mean value of cp on an arbitrary2-dimensional sphere S2(x; r) with center x E R3 and radius r > 0,

(1V)(x,r)4Trr2 f p(y)dy

S2(x,r)

Theorem 4 (Poisson Formula for the Wave Equation). For any two smoothinitial conditions uo, ul : 1R3 - R, the Cauchy initial value problem for thewave equation

1 2

Du =c2 at2 ' u(x,

0)= uo(x), at (x, 0) = ui (x)

has precisely one solution on the space 1R3 x 1R+. Moreover, this solution canbe computed explicitely:

u(x, t) = (t (Iuo)(x, c t)) + t (Iul)(x, c t).

ttII

to+ A (xo, to)

R3

r- (xo, to)

Remark 3. The value u(x, t) of the solution for the wave equation dependsonly on the behavior of the initial values on the boundary of the base of thebackward light cone,

r (xo, to) := {(x, t) E 1R3 x 1R+ : Iix - roil < c It - toe, 0 < t < to} ,

which is a two-dimensional sphere. Therefore, every wave in 1R3 generated byinitial conditions with compact support has a forward as well as a backwardwave-front (Huygens' principle).


Proof of Uniqueness for the solution. We introduce the notation

A- cz1 012

for the wave operator and3

19Uz

Eu(x,t) := 1(a l+(cat Jz

J J)

for the energy. Then the following easy identity holds:

au 3 a au au aE( u)

Now let u be a solution of the wave equation with initial conditions u(x, 0) _0 and au(x, 0)/at = 0, and denote by FT (xo, to) the truncated cone describedby the condition 0 < t < r. By integrating the stated identity, we obtain

a r 3E(u) = 2

Ja au au

at axe t axerr r,

Gauss' theorem (Theorem 27, §3.8) allows us to transform the volume inte-grals into surface integrals:

3

f E(u) ' (N, atj = 2 at axj \N' axjarr er, '=1

The boundary 81'7 consists of three parts. On the first part (t = 0), theenergy E(u) and the derivative au/at vanish; this term does not contributeto either side of the equation. On the second boundary piece (t = r), thenormal vector N is parallel to a/at. Hence the integral on the right-handside of the equation vanishes, whereas the integral on the left-hand side isnon-negative. On the third part of the boundary, the lateral surface of thetruncated cone, we calculate the scalar products

a _xJ. - zo ( a _ C

CN'ax) Ilx - xoll 1 + \N' i+cThus we can transform the difference of the left-hand and the right-handintegrand into an expression which is non-negative, too:

2a au au a c au x) - xo On

E(\N' at / -2

at axi \N' axi / - i + F- cat l I x - xo l l ax)

All in all, the integral formula implies that the energy E(u) vanishes onthe upper boundary (t = r) of the truncated cone E. The height r ofthe truncated cone can be varied, and thus E(u) vanishes identically in the


interior of the cone r- (xo, to). There the function u(x, t) is constant, and,looking at the initial conditions, identically zero.

Proof of the existence of a solution. We calculate the derivatives of thespherical mean with respect to time:

(Iul)(x,ct) = 4nc2t2Jui(y)dy = 41r

f u1(x+ct z)dz,S2(x,Ct) 52

where S2(x, Ct) is the sphere with center x E R3 and radius ct. Denote byS2 the unit sphere. Then Green's formula implies

r C?u l8(lul) s

ax- jat 4w(x, ct) = c J z axj (x + ct. z)dz

2 j=1

2 f N(a), a"1(a)da = l 2 J Aul (y)dy47rct axj ' axJ 41rrt

S2(x.ct) D3(x,ct)

Introducing spherical coordinates (r, a) E (0, oo) x S2 in the three-dimen-sional ball D3(x, ct) and applying the formula d]R3 = r2 dr A da, we obtain

ct

a(at 1) (x, ct) = 4x 2 jr2jAU,(X+ra)da-dr.

This leads to0 S2

Ct

(t (Iul)(x,ct)) = (Iul)(x,ct) + 41rct f r2 fDul(x+ra)da dr,0 S2

and, after further differentiation,2

2(t. (Iu1)(x, ct)) = 4 t

JAu,(x+ct.a)da = c20(t' (Iul)(x, ct))S2

Thus w(x, t) := t- (1ul) (x, ct) is a solution for the wave equation with initialconditions

2

w(x, 0) = 0 = 02 x, 0) and at(x,0) = (Iul)(x,0) = ul(x).

The derivative Ow/8t is also a solution of the wave equation, and it providesthe second summand in the asserted formula.

We have thus solved the wave equation for a scalar function on R3. Theexpression for the vector valued functions E and B follows from a tediousbut routine computation. The explicit formula may be found in Exercise 3.Now we turn to the inhomogeneous wave equation u = f with a smooth


function f : JR3 x R+ -s R, and will solve this for the initial conditionud = 0 = ul. Combining the resulting solution with a solution for thehomogeneous wave equation with arbitrary initial condition constructed inthe preceding theorem, we will arrive at the solution for the inhomogeneouswave equation Du = f with arbitrary initial condition u0, u1. The retardedpotential of a smooth function f : R3 x lR --> JR is defined by the integral

Rf(x,t) := - I f(y,t-r/c) dy41r r

11x-Y11 <d

with the distance function r:= uix - y1l.

Theorem 5. Let f : R3 x R+ - R be a smooth function. Then the re-tarded potential R f is a solution of the Cauchy initial value problem for theinhomogeneous wave equation,

Du = f, u(x,0) = 0 = 8 (x,0).

Proof. Passing to spherical coordinates with center x E R3, we can repre-sent the retarded potential as

fJRf(x,t) _s2

Introducing the parameter r := t - 1: and the function

p(x,t,r) := t47rJ

32

we obtain from p(x, t, t) = 0 the following formulas for the retarded potentialand its derivatives:

tr tr

Rf(x,t) =-c2J p(x,t,r)dr, 8t(Rf)(x,t) = -c2J (x,t.r)dr,0 0

2

5jZ(Rf)(x,t) = -c2 J ate(x,t,r)dr-c2L(x,t,t).0

Calculating the derivative of p with respect to t yields

5'(x, t, t) = Irf f (x, t) da = f (x, t) ,

S2


and, summarizing, we obtain

t

..a 22(Rf)(x,t) = -c2 (x,t,T)dT-c2 f(x,t)0

Applying the wave operator to the retarded potential leads to the formula

t

D(Rf)(x,t) = f(x,t)-c2 f Dp(x,t,T)dr.0

If the parameter T is fixed, the function

p(x, t, T0) = t 4i 0 f f (x + c(t - TO)a, T0) d

S2

is a solution of the homogeneous wave equation. This immediately followsfrom the preceding theorem, since p(x, t, TO) is defined as a spherical mean.Hence the retarded potential solves the inhomogeneous wave equation. 0

The inhomogeneous wave equation Du = f may be interpreted as follows.For simplicity, suppose that f describes the current generated by a chargedparticle moving along the world line y. At the point A, the particle pro-duces an electromagnetic field which moves at the speed of light along thesurface of the light cone with vertex A. Hence the field in B depends on themotion of the particle in A. In this way, the principle of causality appearsas a mathematical consequence of the inhomogeneous wave equation, andits retarded solutions were first described by the physicists Lienard (1898)and Wiechert (1900).

x


The solution of the two-dimensional wave equation can be constructed start-ing from that of the three-dimensional wave equation. To this end we trans-form the spherical mean of a function ¢(x', x2) depending only on the firsttwo spatial variables into a two-dimensional integral by parametrizing thesphere S2 (x, ct) C R3 through

yl = xl + a1, Y2 = x2 + a2, y3 = x3 + c2t2 - (al )2 - (a2)2,

where (al)2 + (a2)2 < c2t2. A straightforward calculation shows that thevolume form of the sphere is described by the formula

dS2 = t2 _ (1) - (a2)2 dal n dal,

from which we obtain the following expression for the spherical mean:

I (xl x2 t) = 1 1 _(x1 - a1, x2 - a2) dal n dal.29rct J c2t2 - (al)2 - (a2)2

D2(O,d)

We treat the retarded potential of a function only depending on the firsttwo spatial variables and time analogously f (x1, x2, t):

_ C xl - al, x2 - a2, rRf (xl, x2, t)

21r J

f(t - r)2 - (a1)2 - (a2)2

dal nda2ndr .

Inserting the resulting expressions into the formulas of the preceding theo-rem, we obtain a representation of the solution of the inhomogeneous two-dimensional wave equation with prescribed initial conditions uo(xl,x2) andu1(x',x2) as an integral over the interior of a disk in R2. From this weconclude that a wave in R2 generated by initial conditions with compactsupport has a forward front, but it does not have a backward front. Thiseffect is well-known, for example in the case of water waves. Stated differ-ently, this means that Huygens' principle (see Remark 3) does not hold intwo dimensions (in fact, it can be shown to fail in all even dimensions).

Now we turn to the question of which conditions rotating an electromagneticwave lead to another electromagnetic wave. Let an electromagnetic waveE(x, t), B(x, t) be given. We rotate these vector fields in the 2-plane spannedby the two vectors through the angle 1/i(x, t):

E` (x, t) = cos 1Ji(x, t) E(x, t) + sin '(x, t) B(x, t) ,

B'(x, t) = - sin 1'(x, t) E(x, t) + cos?i(x, t) B(x, t) .

We discuss the conditions to be satisfied so that the pair (E', B') becomesan electromagnetic wave. Necessarily, ip has to be a solution of the eikonal

9.4. The Relativistic Formulation of the Maxwell Equations

equation

1

(2

- Ilg'ad('+6)fl2 = 0.c2

8t-)

311

Theorem 6. (E*, B') is an electromagnetic wave if and only if the phasefunction 0 satisfies the following conditions:

(1) (grad(tG), B) = 0 = (grad(tG), E);

00 ' B = -grad(O) x E;(2) -

(3) E = grad(O) x B.c

Proof. The vector fields E and B are divergence-free, and hence the diver-gences of E' and B` are easily calculated:

div(E*) = - sin i' (grad(,O), E) + dcos'. (grad('+G), B) ,

div(B') = - cos lO (grad(t/i), E) - sin O (grad(O), B) .

Thus the vanishing of these divergences div(E') = div(B') = 0 is equivalentto the condition (grad(t1), E) = 0 = (grad(Vi), B). Making use of the formula

curl(f V) = grad(f) x V + f - curl(V),

which generally holds in R3, we write down the Maxwell equations for thepair (E', B'). This leads to the following pair of equations:

sin ,p grad(O) x E - c B l = cos 10 _ - E - grad(O) x B)-6T 5T

-C -Nsin t/' (_grad(1iL) x B + . E) = cos 7[i ( B + grad(t/') x E I

which are equivalent to the second and third conditions of the theorem. 0

9.4. The Relativistic Formulation of the Maxwell Equations

Consider Minkowski space R3.1 with coordinates (x, y, z, t) as well as thepseudo-Riemannian metric of index 1 determined by

g :_ (dx1)2 + (dx2)2 + (dx3)2 - c2dt2,

the so-called Minkowski metric (see Example 35, section 3.11). The 1-formsj1 , dx2, dz3 and c dt form an orthonormal frame. Computing the Hodge


operator * of 183.1 on 2-forms leads to the following result:

*(dx' A dx2) _ -dx3 A c dt, *(dx1 A dx3) = dx2 A c dt,*(dxl A c dt) = dx2 A dx3 , *(dx2 A dx3) = -dx1 A c dt,

*(dx2 A c dt) = -dx1 A dx3 , *(dx3 A c dt) = dx' A dx2 .

If w is a 1-form on 1R3, the Hodge operators * of euclidean space R3 and *of Minkowski space 183.1, respectively, are related by

*w.

In R33 we define the 2-form called field strength form by

F :=and the 1-form called density form by

J := 1 wr-C

Theorem 7. The 2-form F has the following properties:

(1) IIFI12 = IIBI12-IIE112.

(2) (F,*F) = 2(E, B).

(3) The first group of the Maxwell equations is equivalent to

dF = 0.(4) The second group of the Maxwell equations is equivalent to

*d*F =

Proof. We calculate the exterior derivative dF of F in 4-space:

dF =

(*&B)+a( B) Adt

= (d(wE)+ 18(*OtwB))

and see that dF = 0 is equivalent to the pair of equations8

(*wB)dR3 (*wB) = 0 and e (WE) C

5iSimilarly, we obtain

d*F=d(*wE - WB A cdt) = dR3 (*wE) + (*wE) A dt - 0 (wB) A cdt,

and, after a slight reformulation,18*d*F = *dR3 (wB)-c8t(WE)

9.4. The Relativistic Formulation of the Maxwell Equations 313

Summarizing, * d * F = 4a J is equivalent to18

d(*WE) = 47rp and * d(wB) =c 8t (wE) +

4c wj.

Remark 4. The 2-form F is a closed form in ]R3.1. Hence by Poincare'slemma there exists a 1-form A such that

dA = -eF.c

The 1-form A comprises the electric as well as the magnetic potential:

F =c 5i

=dR3.'(A

and thus A = -(e/c)A + e 0 dt.

The isometry group of the Minkowski metric g is 0(3, 1), the Lorentz group.It consists of all linear transformations L :183,1 -+ R3,1 which leave g invari-ant. If we represent g by the matrix

A =

0 -C2

then 0(3, 1) can be identified with the (4 x 4) matrices L satisfying

A.

Its Lie algebra o(3, 1) is spanned by the matrices X with

Lt = 0.Of course, 0(3, 1) contains the 3-dimensional group of rotations of euclideanspace R3 as a subgroup, embedded as the upper left 3 x 3 matrices

{ I.0 1

0 M E 0(3,R)1

and generated by the skew-symmetic matrices L1, L2, L3 described in §6.3,Example 14. The missing part of the Lie algebra o(3, 1) is spanned by thethree elements

1

LB'

00 '

B2 :=0

0

0 0 0 0 0 0

f

0

J '

B3 :=

0 0 G

0

0

0

with commutator relations

[B1, B2]= c2L3, [B1, B3[ _ -c2L2, [B2, B3[ = c2Lg,


[L1, B2] _ -B3, [L1, B3] = B2, [L2, B1] = B3, [L2, B3] _ -B1,

[L3, B1] _ -B2, [L3, B2] = B1, [Li, B,] = 0.In the same way as L 1, L2, L3 generate ordinary rotations, the elementsB1, B2, B3 induce hyperbolic rotations, for example,

cosh(c8) 0 0 sinh(cO)/c

OBI0 1 0 0-

e0 0 1 0

c sinh(c 8) 0 0 cosh(c 8)

The defining condition for L E 0(3, 1) implies that L has determinant 1 or-1. The hyperbolic rotation e8Bi and the time reflection diag(1,1,1, -1)realize these values of the determinant. Hence 0(3, 1) has two connectivitycomponents.

Theorem 8.(1) The inhomogeneous Maxwell equations dF = 0 and bF = 41rJ are

invariant under isometries of the Minskowski metric, the Lorentztransformations.

(2) The homogeneous Maxwell equations dF = 0 and d * F = 0 areinvariant under conformal changes of the metric.

(3) The quantities I IFI I2 and (F, *F) are invariant under Lorentz trans-formations.

Proof. An element eox E 0(3,1) acts on differential forms and vectorsby its differential X E o(3, 1). The 2-form F transforms into its pullbackX*(F). But since pulling back commutes with exterior differentiation, dF =0 is equivalent to d(X*F) = 0, which establishes the Lorentz invarianceof the first Maxwell equation. For the second equation, observe that thevolume form dM of any pseudo-Riemannian manifold M is invariant underisometries of the metric; hence the adjoint of d, the coderivative 0, alsocommutes with pullbacks by isometries. Thus, OF = 47rJ is equivalentto X*(bF) = 47rX*(J) and b(X*F) = 4irX*(J). This shows the Lorentzinvariance of the second Maxwell equation.

Now consider a conformal change of the metric on a pseudo-R.iemannianmanifold M, y = e2f g for some function f. The exterior derivative doesnot depend on the metric; the first Maxwell equation is thus trivially con-formally equivalent. An orthonormal basis el,... , e with dual basis of1-forms Qn transforms into

e-fel,...,e-fe, and efv1,...,efa,,.

If M has even dimension n = 2k, we have the remarkable effect that theHodge operators * and * with respect to the metrics g and g coincide on

9.4. The Relativistic Formulation of the Maxwell Equations 315

differential forms in half the dimension of M. For example,

*(a,A...Aak) = -'ak+1A...Aa2k, *(efaiA...Aefak) = E'efak+1A...Aefa2k

for some irrelevant sign e = f1, so that k factors of of can be cancelledin the action of *, which implies the claim. Its follows for the 2-form F on4-dimensional Minkowski space that d * F = 0 is conformally invariant. 0

The conformal group can be identified with O(4, 2), whis has dimension 15.The fact that only the homogeneous Maxwell equations are conformally in-variant was the main reason why Lorentz invariance has been established asthe fundamental invariance property of electrodynamics. However, the fullconformal invariance is useful when dealing with homogeneous situations.The invariance of the scalar quantities I IF112 and (F, *F) is of practical im-portance, too; it means. for example, that E and B stay orthogonal if theywere orthogonal in some frame of reference.

As a 2-form, F is an element of 112(R3"). There exists a purely algebraic,Lorentz invariant map from 112(R3.') into the space of symmetric (2,0)-tensors. Apart from normalizations and an additional multiple of the metric,the image of F under this map is known as the Maxwell stress tensor, whichwe will now discuss.

So let F be any 2-form on Minkowski space, and e1, ... , e4 an orthonormalbasis, i.e., satisfying g(e;, ej) = ej d;j with 1 = el = s2 = E3 = -64. Wedefine

TF :=

where O denotes the symmetric product of 1-forms (see 5.3). This is anSO(3,1)-invariant, symmetric (2, 0)-tensor whose trace is a multiple of IIF112.In the example at hand, we may choose

el = axl, e2 = axe' e3 = ax3and e4 = .

The 2-form F is given by

F = WE A c dt + Bl dx2 A dx3 - B2 dx' A dx3 + B3 dx' A dx2 .

Hence one computes (x4 := cdt)

TF = (Eicdt - B2dx3 + B3dx2)2 + (E2cdt + B1dx3 - B3dx1)2

+ (E3cdt + Bldx2 - B2dx1)2 - (Eldxl + E2dx2 + E3dxa)24

_: 2 dxÒdxJ.


In order to obtain a trace-free endomorphism, one typically defines as theMaxwell stress tensor the combination

Tit :_ -2TF + (F, F) (dx' C dx' + dx2 ^ dx2 + dx3 ®dx3 - c2dt O dt).

Its importance stems from the fact that its components have a physicalinterpretation. For example, its pure time component

T4'4' = -(11E112 + 1IBI12)c2dt2

is the energy density of the electromagnetic field, and the mixed contribu-tions (i = 1.....3)

T"' = 4(ExB)idx'0,cdt

are the components of the Poynting vector E x B. which has the dimension ofan energy flux density. Since Ta' contains basically the same information asF, it is plausible that one can rewrite the Maxwell equations in terms of TMMsolely. For their derivation, we refer to classical textbooks in eletrodynamics.

The electromagnetic potential A is a real-valued 1-form defined on Minkowskispace. We now view the real numbers as the Lie algebra of the compact one-dimensional group U(1). in which the only quantity describing the electronwhich is relevant in Maxwell's theory is encoded---its electric charge. Pass-ing now to the description of more complex elementary particles which carrynot just electric charge, but in addition other characteristics (color, ... ), wehave to replace the one-dimensional group U(1) by a higher-dimensionalcompact Lie group G. If we also want to include gravitation in the model,as understood in the sense of general relativity, flat Minkowski space hasto be replaced by a pseudo-Riemannian manifold. Combining all this, weare led to define a "generalized electromagnetic field with gravitation" asa 4-tuple (Al, g, G, A) consisting of a pseudo-Riemannian manifold, a Liegroup G, and a 1-form A : T(M) -+ g with values in the Lie algebra gof G. The non-commutativity of the group G will become essential in thesequel. The field strength FA is described by a g-valued 2-form, and to theMaxwell equation in the vacuum there corresponds the so-called Yang-Millsequation,

FA := dA + 2 [A, A], DA(* FA) := d * FA + [A, *FA] = 0 .

The possibly complicated topological structure of the base space Al maylead to a situation in which the generalized electromagnetic potential Ais not a globally defined 1-form on M. This leads to topics studied in thedifferential geometry of principal fiber bundles and the theory of connections.

9.5. The Lorentz Force 317

With their help, the models discussed in the physics of elementary particlescan be formulated and studied rigorously.2

9.5. The Lorentz Force

In this section, we discuss the equation of motion for the Lagrangian of amassive particle in an electromagnetic field as introduced in Example 11of Chapter 7. We show that it is a natural generalization of the Lorentzequation, and discuss its solution for a few electromagnetic fields, i.e., atime-independent homogeneous field and a Dirac monopole. Frenet's for-mulation of curve theory will prove to be a suitable tool for the descriptionof the particle's motion.

Let (Mm, g) be a pseudo-Riemannian manifold, A a 1-form on it, and con-sider the Lagrange function

L(v) = 2g(v, v) - A(v) .

Theorem 9 (Generalized Lorentz equation). A curve e(s) : R - Mm is amotion of the Lagrange system (Mm, L) if and only if it satisfies

mds = V(eJdA),

where the right hand side is meant to denote the vector field associated withthe 1 form e J dA relative to the Minkowski metric.

Proof. We use local coordinates x1, ... , xm on Mm. The Lagrangian canthen be rewritten as

L(xl,...,xm,2....... m)m

2 i,j=1 i=1

By definition, e(s) : ]R -p All is a motion of the Lagrange system (Mm, L) ifand only if it fulfills for all k between 1 and m the Euler-Lagrange equation

ds (8i (e(9))) = a k (O(s)).

Since 6(s) = (dx1 /ds, ... , dxm/ds), the left and right sides can be computedto be

OLW M =

m No j dxi dxj 8A; dxi8xk 2 8xk ds ds 8xk ds

to t

2See Thomas Friedrich, Dirac Operators in Riemannian Geometry, Grad. Studies in Math.vol. 25. AMS, Providence, 2000.


and

d aL agile dxi dx' d2xi aAk dxids

(aik (9(s)) I = m>2 axle ds ds + m>9ik ds2 axi dsi

Equating them and rearranging some terms, we obtain the equation

(*)[aAk aAi dxaxi - axk ] ds

e m9ikd2xi +m r99ikds2 ax;

i i,j

1 agi; dx' dx'2 axk I ds ds

Hence it remains to interpret both sides geometrically, as done in the formu-lation of the theorem. We begin by recalling the definition of the covariantderivative as discussed in §5.7, Definition 21. From there, we know that thecovariant derivative may be expressed in terms of the Christoffel symbolsassociated to the chosen local coordinates as

ye _ d2xi a dxi dx' aMds = mE d82 axi + m ds ds axi

But the Christoffel symbols can by computed from the metric (§5.7, Theorem42 ff.):

k - 1 [: km agim a97m 89iir+ 2 m

9 8xk + Oxi - Oxm

hence one checks that the right-hand side of the equation of motion (*) isequal to

r ve a lg 'n ds ' axle

Similarly,

implies that

dA = ez dxi A dxxij

8 aAk aAi dxidA (e,

axk) i axi axk, dwhich is just the left-hand side of (*). We conclude that equationequivalent to

g8

/= dA (k, a

1md,8xk) axk

Since this has to hold for all k, the claim is proved. 0


The most important situation where this result can be applied is the casethat Mm is the flat Minkowski space R3.1 with its pseudo-Riemannian met-ric. As 1-form on R3'1, we choose the form A introduced in the previoussection, which is composed of the magnetic 1-form A on the euclidean spaceR3 and the electric potential ¢:

A = -eA+ecdt = -e(A1dx1+A2dx2+A3dx3)+eodt,

and it satisfies dA = (-e/c) F. We emphasize that time t is viewed as thefourth coordinate of space, whereas the curve parameter s of p is interpretedas the proper time of the particle. The curve p(s) = (x1(8), x2(s), x3(s), t(s))is said to be parametrized in proper time if its tangent vector is normalizedto length -c2,

d - 1 1

2:= c2

In classical mechanics as discussed in Chapter 7 (the Newtonian limit), allvelocities were so small compared to the velocity of light c that it was safe toidentify s with time. We shall henceforth abstain from denoting derivativeswith dots or primes, in order to avoid confusion. By velocities, we mean thetime derivatives vi := dx`/dt with euclidean length v2 := vi + v2 + v32. Theparameters s and t are related through the relativistic -y-factor. To computeit, we express the length of do(s)/ds as

2

=

3 [i)2-c2

[]2 =3 dxt dt 2-c2 [dt]2

= (v2 -c2) dt 2o I I ` ds ds [ dt ds]

ds [ds]dand deduce from its length normalization that

dt

=1 _ ds

ds 1 - v2/c2. 7, and

dt= 1/ry .

Thus, it is comfortable to rewrite do/ds as

do r a a a alds - Y v1ax1 +V2

57X2+v3a7X3 + at

The right-hand side of the generalized Lorentz equation now reads, usingthe definition of F,

WS J c TSj F = +-_r [_(v. E)cdt + (cE1 + (v x B)i)dx`

where v E denotes the euclidean scalar product and v x B the vector productin R3. Its dual vector is

V(doJdA) = ry +(eE;+e(vxB)i) ads c2 at c ax'


Since the left-hand side of the generalized Lorentz equation is

ddg _ ddpmd-ds = mrydt ds

we conclude that its spacelike components yield after simplification the clas-sical Lorentz equation

( 1 ) mdt(ryv) = e E+ _v x B,

and its time component leads to mdry/dt = +(e/c2)v E. This equationdescribes in fact the change in time of the kinetic energy e = mc2-y of theparticle,

(2) t = e(v E).This proves that only the electric field exercises power on the particle, asclaimed in Example 13 of 7.5.

We shall now describe the motion of a charged particle in some specialelectromagnetic fields. Consider the case of a vanishing electric field (E = 0).Equation (2) then implies that the kinetic energy is constant; hence the-y-factor and, in particular, the total velocity v2 are constant, too. Thespatial part of the particle's motion e(s) = (x' (s), x2(s), x3(s)) satisfiesIldWds112 = v2 = const; hence the reparametrized curve q(s) := o(s/v) isgiven in its natural parametrization. The Lorentz equation for this curvecan be written as

mryv2 = cdd

q x B ,

and it is natural to use Frenet's frame to describe the curve q(s) in euclideanspace R3. It is safe again to denote derivatives with respect to the naturalparameter s by dots.

Example 3 (Static and homogeneous magnetic field). Suppose that themagnetic field B is static and homogeneous, i. e., constant in space andtime. Recall that t, h, b denote the tangent, the principal normal and thebinormal vector, respectively, and that c and r are the curvature and thetorsion of q. The Lorentz equation is thus geometrically equivalent to

e-.-txB.c

We differentiate this equation with respect to s and use Frenet's formulasfor the derivative of h and t:

mryv(id +/c(-Kt+'rb)) = eKh' x B.C


Since the vectors t, h', 6 constitute, at every point, an orthonormal frame, weconclude that k = 0. We compute the value of the curvature:

K = 110 =e

jI x BII = eIIBII sin(i),B) = eIIBII 1 - (4.B) 1IIBIIm-yvc mryvc m-yvc

In fact, on can check directly that (4, B) has to be constant, for its derivativeis 2 (ii, B); but by the Lorentz equation, ij is perpendicular to B. Alterna-tively, we could also have derived it from Noether's theorem. In order todetermine the curve's torsion, we first compute its binormal vector,

b = Fxh = xij,

as well as its derivative,

evcq x (h x B) =

y emVc(h(t' B) -

B(h, ryB) h'.b = - x 11 = ymK

The torsion is then

h) _ -e(t' B)rymvc '

and again constant. From Chapter 5, Exercice 7 we know that a curve whosetorsion and curvature are constant is necessarily a helix, the direction ofwhich is given by B. In the special case that (F, B) = 0, the curve is a circlein the plane perpendicular to B with radius of curvature r and synchrotronfrequency w:

1 mryvc v _ eliBIIr =K

=eIIBII'

= r rymc

Example 4 (Dirac monopole3). Although their physical existence couldnever be established, Dirac monopoles have proved important models intheoretical physics. A monopole at the origin of R3 induces the field

c

=B

2eIIiiII3'1

The Lorentz equation is thus

(*)ij

=27mvIMII3J7x g

We are going to prove that the particle moves along a geodesic of a conewith vertex the origin. As in the integration of the geodesic flow or theabstract discussion of Lagrange systems, the proof makes crucial use of aclever invariant of motion. But the invariant of the Dirac monopole differsfrom all invariants encountered so far in the fact that it is not linear in ;

3This discussion has been published by Katharina Habermann (formerly Neitzke) in herarticle K. Neitkze, Die Lorentz-Kraft auf pseudo-Riemannschen Mannigfaltigkeiten, Math. Nachr.149 (1990), 183-214.


hence its existence does not follow from Noether's theorem, but only froma direct calculation. First, we observe that

d !n, (n, 7i) + (n, n) = 0 + 1,

because n is perpendicular to 6j by the Lorentz equation (*), and so thereexists a constant a E R with (n, rl) = s +a. It also implies that the followingderivative vanishes:

ds 0,

which yields the invariant we were looking for. By the Cauchy-Schwarzinequality, it has to be non-negative; hence there exists a constant k > 0such that

Ilnl12 - (n, )2 = k2 and 11,7112 = k2+(S+ a)2.

A short calculation for the curvature of the particle's curve of motion yields

K = Itrill = 1117 x ells = sin(n,il) = 1 -oos (2, _

2TmvJ lot I 2'rmvlIn11s

27mvI InI

1-ln,n>2/1117112 __ 111711'-117,17)2 __ k2 ymvllnl l2 2'rrmll,II3 2-tmvl ln113

If k = 0, the curvature vanishes and the curve is a straight line, i. e., thereexist vectors v", w' E R3 such that n(s) = s v + v7. The equation of motion(*) then implies v' x u7 = 0, i. e., v' and 0 are linearly dependent, so thet wefinally obtain

n(s) = (s + A)t7, A E It ,

which is the equation of a line through the origin. Let us assume from nowon that k > 0. We shall derive the expression for the torsion of the curve.By Frenet's formulas, the principal normal vector hh is

h=-=- 1k7xil.

The binormal vector then becomes

b = ,7xh = -i4x(rlx1),with derivative

dsb = -kn x (n x ii) = n))2kmv7II,II3

Using Grassmann's identity u7 x (u' x ) = u (v, w-) - v' (u, m'), one shows that6 x (u" x (i x ii)) = - (u, v3) i x *7, so that the former expression can further


be simplified to

d g _ (n, n)

(-k__

ds 2mvry11i1113r) X rl 2mvryIIi11I3h.

The torsion is thus equal to

tail) _ _ s+ar = (b, h) _ -2mvryll711132mvry k

+(a+a)23

In particular, the quotient of torsion and curvature satisfies the simple iden-tity r/ic = -(s + a)/k. We prove next that the curve lies on a cone, that is,that there exists a constant vector ii whose angle p with 71 is also constant.

Define the functionss+a _ 1

f(s)

_

2km yv k + (a + a) ' g(s) 2m yv k + -(s+ a)2

They are primitives of K and -T and have the same quotient,

d f (s) dg(s) f (s) _ kds = K' ds g(s)

r

Frenet's formulas thus imply immediately that the vector

iZ :=

is constant. To compute its angle V with rl, we derive an exact expressionfor q by applying Grassmann's identity to the binormal,

1 1b = -krl x (71 x n) _ -k [rl -il(s+a)], q = (s+a)t-k6,

and remark that ii can be rewritten 11= rl/2mkryvllr7l l + h. For the openingangle of the cone we obtain

cos() = (fi, u') - 1

110 141 1 + (2mkyv)


By definition, a curve is a geodesic on a given surface if and only if itsprincipal normal vector is orthogonal to the tangent plane of the surface atany of its points. Denoting by 7P the angle between h and i , we get

(h, u) - 1 _ 2mkvy

Iluli 1 + (2mkvy)2

and we see that this is equal to 1 - cos2(cp) = sin(e). Hence, W and V arerelated through V-V = it/2, and we conclude that h is indeed perpendicularto the tangent plane of the curve. The particle moves along a geodesic ofthe cone, as claimed.

Exercises 325

Exercises

1 (Kirchhoff Formula). Let u(x, t) be a function defined on JR3 x Ht andS C JR3 a compact domain with smooth boundary. Prove, for each point(Xo, to) E SZ x JR, the Kirchhoff formula:

u(1l, to - r/c) r f au (or, to -u(xo, to) = J r dy + J IL r 8Nsl

Ou(a,to- or or(a,to) 8r-1(a,to)+ cr 8t c eN - u(a'to

C OM

where r := Ilxo - yII is the distance to the spatial point xo E R3, N isthe normal vector to the boundary 00, and = Ox - 1/c28it is the waveoperator. Deduce from this the solution formulas for the Cauchy problemof the wave equation by choosing for H a 3-dimensional ball.

2. Under the assumptions of Helmholtz' theorem, the electric and the mag-netic field E and B, as well as the current density field J can be written asthe sum of a divergence-free and a curl-free vector field,

E = Ediv + Ecuri, B = Bdiv + Bcuri and J = Jdiv + Jcurla) Prove that the Maxwell equations can then be written as follows:

18Bdivcurl(Ed;v) = - 8t

chv(Bcuri) = 0

curl(Bd;Y) =1

c at8Ed;v

+47r

Jai,,, div(Ecuri) = 47r LO.

The continuity equation then becomes

OLOdiv(Jcuri) + = 0.

Hint: In the proof of Theorem 2, we proved that, under the assumptionsmade here, any vector field which is at the same time divergence- andcurl-free, has to vanish identically.

b) On the other hand, for the electric field E we already know the decom-position

E _ c +grad(-O).


Prove that this is precisely the Helmholtz decomposition of E if themagnetic potential satisfies the condition called Coulomb gauge:

div(A) = 0.

3. Deduce from Theorem 4 that if Eo(x) and Bo(x) are C3-functions on R3such that divE0 = divB0 = 0, then the Cauchy initial value problem for anelectromagnetic wave

curl(B) = c 5 , curl(E) c at ,

div(B) = 0, div(E) = 0, E(x,O) = Eo(x), B(x, 0) = Bo(x)has the unique solution

E(x, ct)4act f curl Bo(y)dy +

4ac 8t f Eo(y)dy ,

S2(x,d) S2(x,d)

B(x, ct)4act f curl Eo(y)dy + 41

8t f Bo(y)dyS2(x,d) 1S2(x,ct)

4. Using Theorems 1 and 2, determine the electric and the magnetic fieldin 1R3 which is generated in the following situations:

a) a homogeneously charged ball of radius R with constant charge densityP;

b) a charged spherical shell of radius R with constant surface charge densitya;

c) an infinitely extended straight wire of radius R through which a currentwith constant current density j flows.

5. Describe the solution of the classical Lorentz equation for a homogeneous,static electric field (E = const) and vanishing magnetic field (B = 0). Whathappens in the non-relativistic limit of small velocities v << c?

6. The radiation rate of a Lorentz electron is proportional to the secondderivative of the vector p(s) E R3,1:

R211

Compute the general expression for R and discuss the case of planar circularsynchrotron radiation.

Exercises 327

7. Show that the Lie algebra o(3,1) of the Lorentz group is not simple-moreprecisely, that it is the sum of two 3-dimensional ideals.

8 (Infinitely Extended String with Large Oscillations). Consider the Cauchyproblem for the one-dimensional wave equation determined by two functionsup(x) E C2(R), u1(x) E C1(It):

attu = c2atxU, t > 0,

u(x,0) = uo(x), atu(x,0) = u1(x)

Part 1. General Shape of the Solution. Prove that there exist two C2-functions f and g satisfying

u(x,t) = f(x+ct)+g(x-ct).The solution is hence the superposition of a wave traveling to the left andone traveling to the right. Hint: Introduce the new coordinates xt = x ± ctand show that the wave equation is equivalent to the differential equation

ax+(9x- -0.

Part 2. Solution of the Cauchy Problem. With the above Ansatz for thesolution, the following relations have to hold:

uo(x) = f(x) + g(x), u1 (x) = c(f'(x) - g'(x))

By integration of the second equation over the interval [0, x], prove that thegeneral solution has to be given by

Edu(x,t) = 2[uo(x + ct) + ua(x - ct)) + 2cu1(s)ds.

In particular, this argument shows the uniqueness of the solution. How canthe result be understood qualitatively by means of the light-cone ?

a2u


9 (One-sided Infinitely Extended Oscillating String). An oscillating stringextending infinitely in the positive x-direction is modeled by the one-dimen-sional wave equation,

attu = c2axxu, t > 0, x > 0.

In addition to the initial conditions,

u(x, O) = uo(x), 8tu(x, 0) = u, (x),

one has to impose boundary condition which describe the behavior of the"wall" at x = 0 for all times t:

u(0, t) = cp(t).

Here uo, 4o E C2(R+) and ul E C'(R+) are required. Prove (using a similarAnsatz as in the preceding exercise) that the solution is

u(x,t) =

x+d- ui(s)ds,[uo(x + ct) + uo(x - ct)] + -L2cjx d

2 [uo(x + Ct) - up(ct - x)] + p(t - x/C) + ZJd+x

ul (s) ds,d-x

where the upper line is to be taken for points (x, t) in the region I, i. e. belowthe line x = ct, and the lower line for points (x, t) in the region II, abovethe line x = ct. What can be said concerning the behavior of the light-cone, in particular in II? Describe, moreover, the regularity properties ofthe solution on the line x = ct, and explain carefully under which additionalconditions it is of class C2 there.

x=ct

/1

I

x

10 (Oscillating String Fixed on Either Side). For the wave equation onbounded domains a separation Ansatz going back to Bernoulli proved to besuccessful, in that it reduces the problem to one for Fourier series. We arelooking for a solution of the one-dimensional wave equation on the interval[0, 1],

attu = c28xxu, t>0, 0<x<1.The initial and boundary conditions are

u(x, 0) = uo(x), atu(x, 0) = ul (x), u(0, t) = 0, u(1, t) = 0.

Exercises 329

The explanation for the Fourier method and the related regularity propertiesto be required of uo and u1 will be treated separately in the next exercise.Start from the Ansatz

u(x, t) = T(t) X(x) .

a) Prove that there exists a constant A, such that

X" _ T" _X c2T

b) Show that the cases A = 0 and A > 0 necessarily lead to the trivialsolution u = 0; in the case A < 0 there has to exist a natural number ksuch that A = -k27r2 and

uk(x, t) = Xk(x)Tk(t) = sin kirx (ak sin klrct + bk cos kirct) .

The general solution is then obtained as the series00

u(x, t) _ uk(x, t) _ E sin kirx (ak sin k7rct + bk cos klrct) .k=1 k=1

c) Find an integral formula for the coefficients ak, bk depending on theinitial conditions uo and u1. Solution:

2 1 1

uo(x) sin k7rx dx .

ak loreul (x) sin k7rx dx, bk = 2 fo

0

11 (Validity of the Fourier Method). By a theorem of Dirichlet, the Fourierseries of a function f E C1([O,1]) satisfying f (0) = f (1) = 0 is uniformlyconvergent and tends pointwise to f. Prove the following lemma:

For a function f E Ck([O,1]) such that f(0), ..., f(k-1) vanish at 0 and 1,there exists a constant A for which the Fourier coefficients of f satisfy thefollowing inequality:

j f(x)sinn7rxdx< AWk

Formulate necessary conditions for uo and u1, under which the solutionu(x, t) constructed in the preceding exercise is twice differentiable with re-spect to x and t and satisfies the initial conditions.

12 (Inhomogeneous Wave Equation).

Part 1. The inhomogeneous wave equation is understood to be the equation

(*) attu = c28x2u + f(x, t)


with a given function f. We consider this equation on the interval [0, 1] withthe initial conditions

u(x,0) = uo(x), i31u(x,0) = ni(x)

as well as, first, trivial boundary conditions,

u(O,t) = 0, u(l,t) = 0.

Show that the Ansatz

u(x, t) _ an (t) sin(nirx)n=I

satisfies the boundary conditions, and prove that the functions an(t) areuniquely determined. To do so, prove that they solve an ordinary differen-tial equation of second order, and that the initial values an(0),a'n(0) can becomputed from the initial conditions. Hint: The Fourier coefficier..ts of uo,ul and f play an important role here.

Part 2. We are looking for a solution u(x, t) of the inhomogeneous waveequation (*) with arbitrarily given boundary conditions,

u(0, t) = spo(t), u(1, t) = VIM-

Show that the solution of this equation can be reduced to the solution ofan inhomogeneous wave equation with trivial boundary conditions (anda different inhomogeneity f!). Hint: Let 4(x, t) be any function in twovariables satisfying 4(0,t) = spo(t) and fi(l, t) = spi(t), e. g. 4-(x, t) _(1 - x)Wo(t) +xW1(t). Then consider the function v = u - 4D.

13 (Hadamard Example). This exercise intends to illustrate that the Cauchyproblem is not posed correctly for every differential equation, i. e., the so-lution does not necessarily depend "continuously" on the initial conditions.Consider the following Cauchy problem for the Laplace equation: Daterminethe solution of the differential equation

d2u d2u- = 0dt2

+dx2

which at t = 0 satisfy the conditions

u(0,x) = 0, dJu(0.x) = n sinnx,

where k and n are positive integers. Suppose, moreover, that x JR andt>0.

Exercises 331

a) Verify that this problem has the solutionent - e-nt

u(t, x) = 2 nk+l sin nx.

How do (8tu(0, x) I and u(t, x) behave for arbitrarily small t, if n is suf-ficiently large?

b) Now we suppose that we have found the solution 2i(t, x) for the initialconditions

u(0, x) = uo(x), 8tu(0, x) = u1 (x).

What is then the solution of the Cauchy problem with the initial con-ditions

1U(O, x) = uO(x), 8tu(0, x) = u1(x) + sinnx?

Conclude that the Cauchy problem for the two-dimensional Laplaceequation is ill-posed.

Bibliography

Textbooks on Analysis on Manifolds

M. P. do Carmo, Differential forms and applications, Universitext, Springer,Berlin, 1994.

H. Flanders, Differential forms. With applications to the physical sciences,Academic Press, New York-London, 1963.

0. Forster, Analysis 3. Integralrechnung im Rn mit Anwendungen, Vieweg-StudiuYn, Bd. 52, 2. Auflage, Vieweg-Verlag, Braunschweig-Wiesbaden, 1989.

K. Maurin, Analysis 11, Reidel/PWN, Warsaw, 1980.

W. Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill, NewYork, 1976.

M. Spivak, Calculus on manifolds, Addison-Wesley, Reading, MA, 1965.

Textbooks on Differential Geometry

W. Blaschke, H. Reichardt, Einfiihrung in die Differentialgeometrie, Grundl.der math. Wiss., Bd. 58, Springer-Verlag, Berlin, 1960.

M. P. do Carmo, Riemannian geometry, Birkhauser, Basel, 1992.

M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall,Englewood Cliffs, NJ, 1976.

C. Godbillon, Geometrie differentielle et mecanique analytique, Hermann,Paris, 1969.

333

334 Bibliography

A. Gray, Modern differential geometry of curves and surfaces with. Mathe-matics, 2nd ed., CRC Press, Boca Raton, FL, 1998.

M. Spivak, Differential geometry I- V, Publish or Perish Inc., since 1975.

K. Strubecker, Differentialgeometrie I-III, Sammlung Goschen, W. deGruyter Verlag, Berlin, 1969.

R. Sulanke, P. Wintgen, Differentialgeometrie and Faserbi ndel, VEB Dt.Verlag der Wiss., Berlin, 1972.

A. Svec, Global differential geometry of surfaces, VEB Dt. Verlag der Wiss.,Berlin, 1981.

Textbooks on Lie Groups and Lie Algebras

J. F. Adams, Lectures on Lie groups, Benjamin, New York, and Univ. ofChicago Press, Chicago, IL, 1969.

Th. Brocker, T. torn Dieck, Representations of compact Lie groups, GTM98, Springer-Verlag, Berlin, 1985.

J. J. Duistermaat, J. A. C. Kolk, Lie groups, Universitext, Springer, Berlin,2000.

W. Fulton, J. Harris, Representation theory - a first course, GTM 129,Springer, Berlin, 1991.

Textbooks on Symplectic Geometry and Mechanics

V. I. Arnold, Mathematical methods of classical mechanics, GTM 60. Spring-er, Berlin, 1989.

A. T. Fomenko, Symplectic geometry, Advanced Studies in ContemporaryMathematics 5, Gordon and Breach Publ., Amsterdam, 1995.

L. D. Landau, E. M. Lifschitz, Course of theoretical physics, vol. I: Mechan-ics, Pergamon Press, Oxford, 3rd corr. ed., 1994.

A. Sommerfeld, Lectures on theoretical physics, vol. I. Mechanics, AcademicPress, New York, 1952.

W. Thirring, A course in mathematical physics, vol. 1: Classical dynamicalsystems, Springer, Berlin, 1978.

Bibliography 335

Textbooks on Statistical Mechanics and Thermodynamics

R. Becker, Theorie der Wdrme, Heidelb. Taschenb. Bd. 10, Springer-Verlag,Berlin, 1985.

L. D. Landau, E. M. Lifschitz, Course of theoretical physics, vol V. Statisticalphysics, Pergamon Press, Oxford, 3rd rev. ed., 1996.

A. Sommerfeld, Lectures on theoretical physics, vol. V. Thermodynamicsand statistical mechanics, Academic Press, New York, 1956.

D. N. Zubarev, V. Morozov, G. Roepke, Statistical mechanics of non-equili-brium processes, vols. 1 and 2, Akademie-Verlag, Berlin, 1996, 1997.

Textbooks on Electrodynamics

L. D. Landau, E. M. Lifschitz, Course of theoretical physics, vol. II: Theclassical theory of fields, 4th rev. ed., Butterworth-Heinemann, Oxford,1996.

A. Sommerfeld, Lectures on theoretical physics, vol. III: Electrodynamics,Academic Press, New York, 1952.

A. Sommerfeld, Partial differential equations in physics, Academic Press,New York, 1949.

W. Thirring, A course in mathematical physics, vol. 2: Classical field theory,Springer, Berlin, 1986.

SymbolsJ ................ 4, 4,68

* .................7, 7,23

[, ] ..................91

{, ) ................235

O ..................148

V ..................149

0/dt ............... 188

0 ................27,27,53

0/0xi ..............12

0(f) ....... 24, 65, 1010(f) ...............306

Aff(K') ............208Ad .................223

Ad* ................230

ad := Ad........... 224

bid .................151b(s) ............... 132

Bk(U) .............. 19ck ...................26

d ................ 16,685 ...................101

dA, dQ ............ 284div(V)........... 24,24,63

D"(R) .............. 38dMk ................70dM2m ............. 229dV ...................6

Ek .................112

Exp. ...............192

f*., .............11,11,56f*(wk) ..........14,14,67

..................88

9ij, 9ii .............. 59

GF ................231

g .................. 209

GL(n,K)...........208

grad(f) .........23,23,60

grad(V)............ 255

I' .................185

h(s) ...............132

HkR(U) ............ 19

Hk ..................52

El ..................221

712 .................148

Ik ...................27

I(A) ............... 276IV ................. 305

I ................... 148

11 .................. 150

7(p) ............... 275K*(x) ............. 196

K(z, M) ............ 37c(s) ............... 132

Lg .................208

Ev ..................89

C(v) ............... 260

L*(wk) .............. 4

Mk .................47

(M2m, w) .......... 229(M2m, w, H) ....... 236mi(f ), i = 1, 2, 3 ... 164µt ..................272

N(x) ............... 74

0,0 ...............69

0*(F) ............. 231O(n, K) ....... 106, 221

0(t2) .............. 215Wk ...................1

*wy ................23

wi1 .................142

wo................. 232

52 .................. 142

R9 ................. 208

Rf(x,t)............308

curl(V) .............25R(U, V)W ......... 151SL(n, R).. 106, 211, 221SO(n, K) ......106,106,221SU(1,1)............227SU(n) ..............221

s-grad(H) .......... 233supp(ip) ............. 76

S(f?, IA) ............ 276ai ..................141

t( s) ................ 132

TpRn ................11

TXMk ...............54

TMk ................55

T*Mk ............. 230r(s) ................133

U(n) ...............221

V(p) ............12,12,56

vol(Mm) ............ 78X(M2) ............. 165Zk(U) .............. 19Z(B) .............. 278Zc ................ 223

337

Index

action coordinates, 247action integral, 259adiabatic, 290adjoint

minimal surfaces, 171operator, 101representation, 223, 224, 230

Aff(K^), 208affine group, 208, 233, 265Aharonov-Bohm effect, 297angle coordinates, 247angle-preserving map, 108, 181angular momentum, 268area-preserving map, 182Arnold-Liouville theorem, 242atlas, 48

barometric formula, 282Bianchi identity

first, 186second, 191

binormal vector, 132Biot-Savart law, 303Birkhoff's ergodicity theorem, 235black-body radiation, 293Bohr-Sommerfeld condition, 248Boltzmann distribution, 281Boltzmnann's constant, 285boundary

of a chain, 27of a manifold, 53

Bour's minimal surface, 201Brouwer's theorem, 40

canonicalcoordinates, 236

ensemble, 280symplectic structure, 230

Carathdodory construction, 79

Carnot cycle, 290catenary, 172, 198catenoid, 105, 172, 201Cauchy problem

for the Laplace equation, 330for the wave equation, 305

Cauchy'sintegral formula, 38theorem, 37

Cauchy-Riemann equations, 37Cayley transformation, 227center, 223central force, 267chart, 48chart transition, 48Christoffel symbols, 174, 185Clairaut's theorem, 178, 262closed, 18coadjoint representation, 230coclosed, 102Codazzi-Mainardi equation, 153commutator, 91completely integrable system, 244configuration space, 253conformal group, 315conformal map, 108, 181conjugation action, 222connection form, 142, 186conservation of energy

for Lagrangian systems, 261for Newtonian systems, 256in statistical mechanics, 273

conservation of information entropy, 277

339

340

continuity equation, 296Coulomb gauge, 326Coulomb potential, 302covariant derivative

on manifolds, 184, 188, 190on surfaces, 149, 153

cubic homology group, 30curl, 25current density vector, 295

curvature

form, 204Gaussian, 156geodesic, 173lines, 201mean, 156normal, 173of a curve, 132, 198principal, 167scalar, 193sectional, 193tensor, 151, 186

cylinder, 158, 168, 176, 183

d'Alembert-Lagrange theorem, 258d'Alembertian, 101Daniell-Stone functional, 79Darboux frame, 172Darboux theorem, 236Darboux vector, 199de Rham cohomology, 19density form, 312density function, 272

of an electric charge, 295diffeomorphism group, 88differential form, 13, 67

closed, 18enclosed, 102exact, 1Sharmonic, 102left-invariant, 210

dimension, 49, 131Dirac monopole, 321Dirichlet problem, 95distribution, 112

integrable, 112

involutive, 113

divergence, 24, 62geometric interpretation, 91

dual 1-form (of a vector field), 23, 72Dulong-Petit rule, 289

effective potential, 267eikonal equation, 311Einstein equation, 204Einstein space, 194energy, 267

for Lagrangian systems, 260

for Newtonian systems, 255free, 279, 283inner, 279, 283kinetic, 254of a statistical state, 273of a thermodynamic system, 284

energy density, 316Enneper's minimal surface, 201entropy, inner, 279, 283, 284equation of state, 279

general, 283equidistribution, 276equilibrium state, 272Erlanger Programm, 207Enter characteristic, 165Enter equations, 267Euler-Lagrange equations, 258exact, 18exponential map, 192, 212exterior

algebra, 3derivative, 15, 68form, Inormal vector field, 74product, 2

Fenchel inequality, 137, 141field

electric, 295magnetic, 295

field strength form, 312first Bianchi identity, 186first fundamental form, 148first integral, 177, 236, 265fixed point, 38fixed point property, 38flow (of a vector field), 88form

differential, 13exterior, 1

Frenet formulas, 133Frenet frame, 132Frobenius' theorem, 113fundamental theorem

of curve theory, 133of surface theory, 143, 153

fundamental theorems ofthermodynamics, 284

Calerkin's method, 41y-factor, 319gas

ideal, 288real, 293

Gauss equation, 153Gauss'

mean value theorem, 99

Index

Index

theorem, 83Gauss-Bonnet formula, 164, 179Gaussian curvature, 156Gay-Lussac experiment, 292geodesic

curvature, 173line, 174, 192spray, 254

Gibbs state, 280Gibbs' fundamental equation, 284GL(n,K), 208gradient, 23. 60

symplectic, 233graph, 49, 145, 148, 159Green's formula

first, 35, 84second, 36, 84

group of diffeomorphisms, 88

Hadamard. J., 330hairy sphere theorem, 82half-space, 52Hamilton equations, 234Hamilton function, 263Hamilton's theorem, 263Hamiltonian quaternions, 221, 226Hamiltonian system, 236harmonic differential form, 102harmonic function, 94

Liouville's theorem, 99maximum principle, 99mean value theorem, 99Poisson formula, 99

heat form, 284hedgehog theorem, 82helicoid, 172, 201helix, 131, 176, 198, 266Helmholtz's theorem, 104, 304Hessian form, 196Hilbert, David, 158Hodge operator, 7, 23Hodge's theorem, 104, 304Hodge-Laplace operator, 101homogeneous space, 220homotopy, 33, 81Hopf's theorem, 85Hopf-Poincare theorem, 166Huygens' principle, 305, 310hyperbolic plane, 148, 158, 179, 190, 202,

227

ideal gas, 288Igel, Satz vom", 82imaginary part surface, 147, 161index

of a curve, 199of a scalar product, 5

of a vector field, 166induced

differential form, 14, 67exterior form, 4

information, 276information entropy

of a probability measure, 276of a statistical state, 277

inner entropy, 279, 283, 284inner product, 4, 68integrability condition, 114integral curve, 87integral manifold, 112integrating factor, 120irreversible process, 277isometry, 180isotherm, 290isothermic coordinates, 170isotropy group, 231

Jacobi, 200Jacobi identity, 93, 235

Kerr metric, 196Kirchhoff formula, 325Kirillov form, 232Klein, Felix, 207

Lagrange function, 257Lagrange's theorem, 4Lagrangian system, 258

hyper-regular, 262Lambert projection, 183Lancret's theorem, 134Laplace-Beltrami operator, 65Laplacian, 24, 65Lebesgue measure, 79left-invariant, 209, 210Legendre transformation, 260Levi-Civita connection, 184Lie algebra, 209Lie derivative, 89Lie group, 208Lie, Sophus, 207line integral, 30Liouville form, 230, 265Liouville's equation, 273Liouville's theorem

for harmonic functions, 99for symplectic manifolds, 234

Lienard, 309Lorentz equation, 266

classical, 320generalized, 317

Lorentz force, 259, 260, 262Lorentz group, 313

341

342

manifoldflat, 188orientable, 69simply connected, 80symplectic, 229with boundary, 52without boundary, 47

Maupertuis-Jacobi principle, 257Maurer-Cartan equations, 210Maurer-Cartan form, 210maximum principle, 99Maxwell distribution, 281, 292Maxwell equations

in classical formulation, 295relativistic version, 312

Maxwell relations, 290Maxwell stress tensor, 315mean curvature, 156Mercator projection, 182microcanonical ensemble, 282minimal surface, 169

Bour's, 201Enneper's, 201

Minkowski space, 101, 311Minkowski-Steiner theorem, 167modular surface, 146, 159, 201module, 222Mobius strip, 51moment map, 239, 268momentum, 267motion, 254

natural parametrization, 131Neumann problem, 95Newton potential, 299Newton's equation, 254Newtonian system, 254

with potential energy, 255Noether's theorem, 177, 238, 261normal curvature vector, 173normal vector field, 74

O(n, R), 106, 221orientation

induced, 75of a manifold, 69of a vector space, 6

osculating plane, 134Ostrogradski formula, 83

parallel displacement, 189parallel vector field, 188parametrized curve, 131partition function, 278, 283partition of unity, 76Peano curve, 129perihelion precession, 252

Pfaffian system, 112phase space, 253Planck's radiation law, 293Poincar4's lemma, 20, 81Poincar6's return theorem, 234point

elliptic, 167flat, 167hyperbolic, 167parabolic, 167umbilic, 167

Poisson bracket, 235Poisson formula

for harmonic functions, 99for the wave equation, 305

polar coordinates, 57potential

chemical, 286electric, 297magnetic, 296Newton's, 299retarded, 308thermodynamic, 289

potential energy, 255Poynting vector, 316pressure, 286principal curvature, 167principal normal vector, 132principle of least action, 259probability current, 275proper time, 319pseudo-Riemannian metric, 100pseudosphere, 107, 158pullback

of a differential form, 14, 67of an exterior form, 4

quaternions, 221, 226

Rayleigh-Jeans law, 294real gas, 293real part surface, 147, 161representation

adjoint, 230

coadjoint, 230, 265defining, 222irreducible, 227of a Lie algebra, 221of a Lie group, 221trivial, 222

retarded potential, 308retarded solutions, 309reversible process, 277Ricci tensor, 193Riemann surface, 106Riemann, Bernhard, 197Riemannian curvature tensor, 186

Index

Index

Riemannian metric, 59

scalar curvature, 193scalar product, 4Schrudinger equation, 297Schwarzschild-Eddington metric, 196, 204second Bianchi identity, 191second fundamental form, 150sectional curvature, 193Shannon, C. E., 276signature (of a scalar product), 5simply connected, 80singular

chain. 27cube, 26

SL(2, R), 106, 211SL(n, K), 221slope line, 134smooth map, 26, 34, 39solid body, 289SO(n, R), 106, 221space form, 196space of constant sectional curvature, 196spherical coordinates, 59spherical mean, 305spherical pendulum, 249standard cube, 27star-shaped set, 19stationary terminal distribution, 272statistical state, 271stereographic projection, 107, 181Stokes' theorem, 32, 79

classical version, 36, 85structural equations

of a Lie group, 210of a manifold, 187of a surface, 133, 142

SU(1,1), 227SU(n), 221support, 76surface of revolution, 144, 148, 158, 177, 185,

200Sylvester's theorem, 4symmetric product, 148symplectic

coordinates, 236diffeomorphism, 238form, 229gradient, 233manifold, 229structure, 229volume form, 229

synchrotron frequency, 321synchrotron radiation, 326system, completely integrable, 244

343

tangent spaceof Rn, 11of a manifold, 54

temperature, 284, 285tensor product (of representations), 227Theorems Egregium, 142, 163thermal efficiency, 291thermodynamic relations, 290Toda lattice, 266torsion, 133, 198torus, 50total differential equation, 121tractrix, 107, 158transitive action, 220

U(n), 221unit cube, 26unit normal vector field, 74

van der Waals equation, 293vector field, 12, 56

complete, 88components of, 57flow of, 88fundamental, 231left-invariant, 209normal, 74parallel, 188related, 93

velocity of light, 296volume, 78volume form, 6, 70, 229

wave (electromagnetic), 304wave equation, 304

Cauchy problem for the, 305Poisson formula for the, 305

wave function, 297wave operator, 101, 306Weierstrass representation, 171Weingarten map, 152, 156Wiechert, 309Wien's law, 294winding form, 18winding number, 137, 199work form, 284

Yang-Mills equation, 316

tangent bundle, 55

From a Review of the German Edition:Drawing on his great experience in research, writing books, teaching, and working withstudents, Friedrich presents once more a clearly written, smoothly readable selfcontained textbook The mathematical material and approaches are well motivated,enriched by valuable considerations and reflections. Proofs are elegant, not too tech-nical and carefully performed ... Each chapter finishes with exercises designed toincrease comprehension ... For any student who has passed the linear algebra courseand calculus, this book offers on excellent opportunity to learn global analysis and itsapplications to mathematical physics.

-Mathematical Reviews

This book introduces the reader to the world of differential forms and theiruses in geometry, analysis. and mathematical physics. It begins with a few basictopics, partly as review, then moves on to vector analysis on manifolds and thestudy of curves and surfaces in 3-space. Lie groups and homogeneous spaces arediscussed, providing the appropriate framework for introducing symmetry inboth mathematical and physical contexts.The final third of the book applies themathematical ideas to important areas of physics: Hamiltonian mechanics, statis-tical mechanics, and electrodynamics.

There are many classroom-tested exercises and examples with excellent figuresthroughoutThe book is ideal as a text for a first course in differential geometry,suitable for advanced undergraduates or graduate students in mathematics orphysics.

\\1ti !.r !h: \\cIIwww.ams.org

Documents

Global Analysis Agricola&Friedrich