Differential Geometry and

tensors

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Differential Geometry and

tensors

K. K. DubeProfessor (Mathematics)

Dept of Mathematics, Statistics and Computer ScienceG. B. Pant University of Agriculture and Technology

Pantnagar – 203145Uttarakhand (India)

I.K. International Publishing House Pvt. Ltd.New DelhI ● BANGAlore

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Published by

I.K. International Publishing house Pvt. ltd.S-25, Green Park extensionUphaar Cinema MarketNew Delhi - 110 016 (India)e-mail: info@ikinternational.com

ISBN: 978-93-80026-58-9

© 2009 I.K. International Publishing house Pvt. ltd.

10 9 8 7 6 5 4 3 2 1

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or any means: electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission from the publisher.

Published by Krishan Makhijani for I.K. International Publishing house Pvt. ltd., S-25, Green Park extension, Uphaar Cinema Market, New Delhi-110 016 and printed by rekha Printers Pvt. ltd., okhla Industrial Area, Phase II, New Delhi-110 020.

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Dedicated to

My Beloved Late Parents

Sri B. P. Dube and Smt. Padmawati Dube

Whose memories had always been an inspiration

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Preface

The aim of this book is to provide a textbook on Differential Geometry and Tensors, which is being taught in the present three years degree course in Mathematics.

The purpose of this book is to give a simple, lucid, rigorous and comprehensive account of fundamental notions of Differential Geometry and Tensors. The book is self contained and divided into two parts, section-A and section-B. Section-A deals with Differential Geometry and section-B devoted to the study of Tensors. Section-A consists of six chapters, dealing with theory of curves, envelopes and developables. Third and fourth chapters deal with curves on surfaces and fundamental magnitudes, curvature of surfaces and lines of curvature (local non-intrinsic properties of surfaces). The fifth chapter is devoted to the study of fundamental equa-tions of surface theory. The six chapter of section-A deals with Geodesics.

Section-B is devoted to the study of Tensors. First introduction is given, then chapter one deals with some preliminaries. Chapter two is devoted to the study of Tensor Algebra and third chapter deals with Tensor Calculus. Fourth chapter is devoted to the study of Christoffel sym-bols and their properties. Fifth chapter deals with riemann symbols and einstein space, and their properties. Physical components of contravariant and covariant vectors is also discussed. Chapter six deals with Geodesics and Parallelism of vectors. The last chapter deals with defini-tion of Differentiable manifolds, Charts, Atlases and some examples.

Numerous examples are either wanted out are given as an exercise in both the sections, to grasp the subject easily and clearly.

I offer my sincere thank to Professor S. B. Pandey, S. S. Jeen Kumaun University Campus, Almora and Dr. ram Niwas Prof. and head, Dept. of Maths, lucknow University, lucknow for their encouragement in writing this book. I shall be failing in our duties if I do not acknowl-edge the help of my wife Smt. Shanti Dube and my sons and daughter in-laws rahul-Sweta and rohit-Manisha for their constant encouragement. I also thank my both the grandsons Mr. Advit Dube and Ahaann Dube for providing me happiness while completing this task.

K. K. Dube

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Contents

Preface vii

Section A: DifferentiAl Geometry

VeCTor NoTATIoNS 3

1. Theory oF CUrVeS IN SPACe 91.1 Curves in Space 91.2 Class of a Function or a Curve 101.3 Tangent line 101.4 order of Contact of Curves and a Surface 161.5 The osculating Plane 181.6 Osculating Plane at a Point of Inflexion 191.7 osculating Plane at a Point of the Curve of Intersection of Surfaces

f1 (x, y, z) = 0, f2(x, y, z) = 0 221.8 Principal Normal Curvature 241.9 Binormal 261.10 Curvature and Torsion of a Curve of Intersection of Two Surfaces 281.11 helices 361.12 Fundamental Theorem for Space Curves and Intrinsic Equations 391.13 osculating Circle (or the Circle of Curvature) 401.14 osculating Sphere or Sphere of Curvature 441.15 Spherical Indicatrix 551.16 Involutes and evolutes 581.17 Bertrand Curves 65

2. eNVeloPeS AND DeVeloPABleS 792.1 Family of Surfaces (one Parameter) 792.2 edge of regression 812.3 Envelope of a System of Surfaces Whose Equation Contains Two Parameters 822.4 ruled Surfaces 87

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x • Contents

2.5 Condition for a Developable and Skew ruled Surfaces 882.6 Developable Surface 892.7 Developable Associated with Space Curve 912.8 Central Point, line of Striction and Parameter of Distribution 100

3. CUrVeS oN SUrFACeS AND FUNDAMeNTAl MAGNITUDeS 1073.1 Introduction 1073.2 Curvilinear Equations of the Curve on Surface 1093.3 First order Magnitudes 1113.4 Direction on a Surface 1133.5 Second order Magnitudes 1163.6 Derivative of n, the Surface Normal 1203.7 Curvilinear Area 1213.8 Direction Coefficients 1223.9 Family of Curves 1243.10 helicoids 129

4. CUrVATUre oF SUrFACeS AND lINeS oF CUrVATUre 133local Non-Intrinsic Properties of Surface 1334.1 Definitions 1334.2 Curvature Vectors 1334.3 Principal Directions, Principal Curvature and lines of Curvature 1384.4 lines of Curvature 1474.5 Third Fundamental Form: The Quadric 1524.6 Conjugate Directions 1624.7 Conjugate Systems 1644.8 Asymptotic lines 1654.9 Null lines 169

5. FUNDAMeNTAl eQUATIoNS oF SUrFACe Theory 1755.1 Gauss’s Formula for r11, r12, r22 1755.2 Gauss Characteristic Equations 1775.3 weingarten Formulae 1785.4 Mainardi-Codazzi Equations 1795.5 Derivative of the Angle 180

6. GeoDeSICS 1856.1 Geodesic Property 1856.2 Equations of Geodesic 1856.3 Geodesic on Developable Surface 1956.4 Torsion of a Geodesic 1966.5 Vector Curvature 197

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Contents • xi

6.6 Geodesic Curvature 1986.7 Curves in relation to Geodesics 2026.8 Geodesic Parallels 2036.9 Geodesic Polar Co-ordinates 2066.10 Gauss-Bonnet Theorem 2086.11 Theorems on Parallels 2116.12 Geodesic ellipses and hyperbolas 2126.13 Conformal Mapping, Geodesic Mapping 2136.14 Fundamental existence Theorem for Surfaces 2156.15 Geodesics on F (x, y, z) = 0 216

Section B: tenSorS

1. INTroDUCTIoN 2351.1 Determinants 2351.2 Differentiation of a Determinant 2381.3 rank of Matrix 2401.4 Linear Equations, Cramer’s Rule 2401.5 linear Transformations 2401.6 Quadratic Form 241

2. TeNSor AlGeBrA 2472.1 Coordinates, Vectors, Tensors 2472.2 Transformation of Coordinates 2482.3 Invariants 2502.4 Contravariant and Covariant Vectors 2512.5 Tensors of order two: “Contravariant Tensor of Second order” 2562.6 Fundamental operations with Tensors 2592.7 relative Tensors or Vectors 267

3. TeNSor CAlCUlUS 2813.1 Introduction 2813.2 riemannian Space Fundamental Tensor 2823.3 Conjugate Tensor of Fundamental Metric Tensor gij or

reciprocal Tensor of gij 2853.4 Associated Tensors 2893.5 length of a Curve, Magnitude of a Vector 2913.6 Unit Vector and Null Vector 2933.7 Angle between Two Non-Null Vectors: orthogonal Vectors 2953.8 euclidean Space of n-Dimension 297

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xii • Contents

4. ChrISToFFel SyMBolS, CoVArIANT DIFFereNTIATIoN, AND TheIr ProPerTIeS 2994.1 Christoffel Symbols, Covariant Differentiation, and Their Properties 2994.2 law of Transformation of Christoffel Symbols 3064.3 Covariant Derivative of Covariant Vectors, and Contravariant Vector 3114.4 Covariant Derivative of Tensors of Type (0, 2), (2, 0) and Mixed Tensors 3144.5 Covariant Differentiation of Sums (Difference) and Product 3204.6 Gradient of Scalar, Divergence of a Contravariant Vector,

Covariant Vector and Conservative Vector 3234.7 laplacian of an Invariant, Curl of a Covariant Vector, Irrotational Vector 327

5. rIeMANN SyMBolS 3335.1 Curvature of riemannian Space 3335.2 Properties of Rijk

l and Rlijk 3365.3 ricci Tensor and Scalar Curvature 3395.4 einstein Space and einstein Tensor 3425.5 Intrinsic Derivative and Physical Components of a Vector Ap or Ap 345

6. GeoDeSICS, rIeMANNIAN CoorDINATeS, GeoDeSIC CoorDINATeS AND PArAllelISM oF VeCTorS 3516.1 Families of Curves, euler's Conditions 3516.2 Geodesics 3536.3 riemannian and Geodesic Coordinates 3546.4 Parallelism of Vectors 357

7. DIFFereNTIABle MANIFolDS AND rIeMANNIAN MANIFolDS 3637.1 Differentiability Class 3637.2 Differentiable Manifolds 364

Index 367

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About the Book

This book has been systematically organized in a easy way to read. This book will surve as suitable text cum reference book for the students of M.Sc. and B.Sc. honours from various universities.

The book is divided into two sections A and B. Section-A consists of six chapters and section-B consists of seven chapters. Section-A deals with Introduction followed by Theory of Curves, envelopes and Developables. Curves on surfaces and fundamental magnitudes, curva-ture of surfaces and lines of curvature. Fundamental equations of surface theory and geodesics and their properties. Section-B deals with Tensors. This part begins with the concepts and techniques. This part deals with tensors of different orders. Einstien summation convention, Kronecker delta symbol and also concepts of tensor algebra and tensor calculus. The author also studied about Riemannian geometry and their properties. Some basic definition of differ-entiable manifold with example is also deal.

About the Author

Dr. K. K. Dube is a Professor in the Department of Mathematics, Statistics and Computer Science at G. B. Pant. University of Agriculture and Technology, Pantnagar (Uttarakhand). Dr. Dube obtained Ph.D degree from lucknow University, lucknow and also recipients of Post-Doctoral fellow (CSIr). he has 30 years of teaching experience of UG/PG classes. he has to his credit more than 71 research papers published in national/international journals of repute.

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1Theory of Curves in Space

Differential Geometry is that branch of mathematics which deals with the space curves and surfaces by means of differential calculus. The properties or relations are derived by means of differential coefficients of the magnitudes which are connected with the curves and surfaces. The subject, therefore, is called differential geometry.

A geometric property which is independent of the geometric configuration of a curve or a sur-face under consideration as a whole but depends only on the form of the configuration in an arbitrarily small neighbourhood of any kind is called local property and the geometry based on this property is called local geometry. The study of curves and surfaces as a whole is called global property and the geometry based on this is called global geometry, but here we shall deal with local properties instead of global one.

1.1 Curves in spaCe

When every part of the curve in space is confined to a plane, it is called a plane curve, otherwise it is said to be skew twisted or tortuous curve. Some authors refer it as skew or twisted curve of double curvature or a curve of torsion or curve of inflexion.

Curves in space will be defined usually by an analytical definition. In such a case, the position vector of any point on the curve in space will be expressed in terms of single parameter, t. Thus, if r is the position vector of any point P on the curve, then equation of the curve is written as

r r t= ( ), (i)

where r t( ) denotes r as a function of t.

In Cartesian coordinates x, y, z, equation (i) may be written as

r xî yj zk= + + (ii)

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10 • Differential Geometry and Tensors

Thus, for a space curve x, y, z are functions of t. Thus, space curves is given by,

x x t y y t z z t= = =( ), ( ), ( ) (iii)

These are called parametric equations of space curve. The positive direction in the direction of increasing t.

Now eliminating t between x = x (t) and y = y (t), and also between y = y (t), and z = z (t), we get

Q1(x, y) = 0 and Q2( y, z) = 0 respectively. (1)

These equations represent cylinders. Thus, above equation (1) represents the curve of section of cylinders given by (1).

As any equation in x, y, z represents a surface, therefore, a space curve may be regarded as the intersection of two surfaces given by the equations

f1(x, y, z) = 0 and f2(x, y, z) = 0 (2)

1.2 Class of a funCtion or a Curve

A real valued function f is called a function of class n or cn-function, defined over a real interval I, if it has continuous n th order derivative.

If f is single valued and has continuous derivatives of all order at each point of the interval, then f is called analytic over that interval I and f is said to be cw-function.

If f is derivable infinitely many times, then it is called c-function or function of class .

A vector valued function r (x, y, z) defined on an interval I is said to be class n, if all components are of class n. Also if dr dt r/ = ≠ 0 on I, then r (t), is said to be a regular function. Thus, the path of class n is a regular function of class n.

equivalent paths

Two curves c1, c2 of the class n on the interval I1, I2 are said to be equivalent if there exists a mapping of I1, on to I2 that transposes c1 to c2.

1.3 tangent line

Let r = r (t) be the given curve C and P be the point on the curve at which tangent line is to be obtained. Let Q be another point on the curve in the neighbourhood of P. Let

OP r t r OQ r t t r r

= = = + = +( ) ( )and δ δ

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Theory of Curves in Space • 11

∴ = −⇒ + −

= + −

⇒ = + −

PQ OQ OP

r r r

r t t r t

r

t

r t t r

( )

( ) ( )

( )

δδ

δδ

δ (( )

lim( ) ( )

t

t

rdr

dt

r t t r t

tt

δδδδ

⇒ = = + −→

0

is nothing but a line along the tangent PT at P, which is the limiting position of the chord PQ as Q → P, i.e., as t → 0.

Now PT passes through P whose position vector is r and which is along the vector dr / dt, there-fore, the vector equation of the tangent PT at P is

R r r= + λ , (1)

where is an arbitrary scalar constant and R is the current position vector of any point on the tangent.

If R X Y Z r x y z r x y z= =( , , ), ( , , ) ( , , )and

Then from (1), we have

Xi Yj Zk xi yj zk xi yj zk

X x x Y y y Z z z

X

+ + = + + +⇒ = + = + = +

⇒

λλ λ λ

( , , )

, ,

−− = − = − =x

x

Y y

y

Z z

z

λ (2)

Thus, from above equation (2), it follows that x y z, , are direction-ratios of the tangent line.

RemaRk

When the curve is given by

f1(x, y, z) = 0 = f2 (x, y, z)

CT

Q

P(t)

r

O

r + δr

Q(t+δt)

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Differential Geometry And Tensors

Publisher : IK International ISBN : 9789380026589 Author : K.K. Dube

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