Part III: Applications of Differential Geometry to

Physics

G. W. Gibbons

D.A.M.T.P.,Cambridge,

Wilberforce Road,

Cambridge CB3 0WA,U.K.

January 20, 2011

1 Action of Groups on Manifolds

1.1 Groups and semi-groups

A group G is a set gi with a multiplication law, G×G→ G which we write multiplica-tively and which satisfies

closure (i) g1g2 ∈ G ∀ g1 , g2 ∈ G (1.1)

associativity (ii) (g1g2)g3 = g1(g2g3) ∀ g1 , g2 , g3 ∈ G (1.2)

identity (iii) ∃e ∈ G s.t. eg = ge = g ∀ g ∈ G (1.3)

inverse (iv) ∀g ∈ G ∃ g−1 ∈ G s.t. g−1g = gg−1. (1.4)

If one omits (iv) one gets a semi-group. These frequently arise in physics when irre-versible processes, such as diffusion are involved.

1.2 Transformation groups: Left and Right Actions

Usually one is interested in Transformation group acting on some space X, typically amanifold, such that there is a map φ : G×X → X such that X ∋ x→ φg(x) and

left action φg1 φg2 = φg1g2, φe(x) = x. (1.5)

Such an action is called a left action of the group G on X or sometimes in physics arealization1. The action of a map ψ would be called a right action if

1Often one speaks of a non-linear realization to distinguish the action from a linear action, linear

realization or representation of G on X . We shall treat the special case of representations shortly.Of

course the word ‘action’has nothing to do with action functionals

1

right action ψg1 ψg2 = ψg2g1. (1.6)

In other words the order of group multiplication is reversed under composition. Ourconventions for composition are that the second map is performed first, i.e

f g(x) = f(g(x)) (1.7)

which coincides with the standard order for matrix multiplication in the case of linearmaps. In terms of arrows, and if we think of in the standard way of reading input tooutput from left to right, we have

X → gX → fX. (1.8)

It would, perhaps, be more logical to read from right to left

X ← fX ← gX. (1.9)

It sometimes happens that one needs to consider right actions. This gives rise tovarious sign changes and so one must be on one’s guard. Of course, given a right actionψ , one may always replace it by a left action provided each group element is replaced byits inverse:

ψg−1

1

ψg−1

2

= ψg−1

2g−1

1

= ψ(g1g2)−1 . (1.10)

1.3 Effectiveness and Transitivity

The action (either left or right) of a transformation group G on X is said to be effectiveif ∄g ∈ G : φg(x) = x∀, , x ∈ X, i.e which moves no point. We shall mainly be interestedin effective actions since if the action of Gis not effective there is a subgroup which is.

The action is transitive if ∀x1 , x2 ∈ X, ∃ g ∈ G : φg(x1) = x2. In other wordsevery pair of points in X may be obtained one form the other by means of a grouptransformation.

If the group element g is unique the action is said to be simply transitive. Otherwisethe action is said to be multiply transitive. subsectionActions of groups on themselvesOne has three important actions

Left action Lh : g → hg, h regarded as moving g (1.11)

Right action Rh : g → gh, h regarded as moving g (1.12)

Conjugation Ch : g → hgh−1, h regarded as moving g (1.13)

Note that the left and right actions are simply transitive and they commute with oneanother

Lg1 Rg2 = Rg2 Lg1 . (1.14)

By contrast, , which is in fact a left action, commutes with neither and fixes the identityelement or origin e of the group G

Ch(e) = e ∀h ∈ G. (1.15)

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Conjugation is a natural symmetry of a group. Roughly speaking, it is analogous tochange of basis for a vector space. Another way to view it is a sort of rotation aboutthe identity element e. Two subgroups H1, H2 ⊂ G are said to be conjugate if there is ag ∈ G such that for all h1 ∈ H1 and h2 ∈ H2, gh1g

−1 = h2, and thus gH1g−1 = H2. Two

subgroups related by conjugation are usually regarded as being essentially identical sinceone may be obtained from the other by means of a ‘rotation’about the identity origin e.A subgroup H is said to invariant or normal if gHg−1 = H , ∀g ∈ G.

1.4 Cosets

If H is a subgroup of G it’s set of right cosets G/H = g ∈ G : g1 ≡ g2 iff ∃H ∈ H :g1 = hg2. It’s set of left cosets H\G = g ∈ G : g1 ≡ g2 iff ∃H ∈ H : g1 = g2h.Our notation, which may not be quite standard is such that right cosets are obtained bydividing on the right and left cosets by dividing on the left. Note that, because right andleft actions commute, Lg the left action of G descends to give a well defined action on aright coset and the right action Rg descends to give a well defined action on a left coset.

1.5 Orbits and Stabilizers

The orbit OrbG(x) of a transformation group G acting on X is the set of points attainablefrom x by a transformation, that is OrbG(x) = y ∈ X : ∃ g ∈ G : φg(x) = y. Clearly Gacts transitively on each of its orbits. If G acts transitively on X, then X = OrbG(x) ∀x ∈X. One may think of the set of right cosets G/H as the orbits in G of H under the leftaction of G on itself and the right cosets H\G as the orbits of H under the right actionof G on itself.

If G acts multiply transitively on X then H is said to be a homogeneous space and∀x ∈ X, there is a subgroup Hx ⊂ G each of whose elements h ∈ Hx fixes or stabilizesx, i.e. φh(x) = x∀xinHx. The subgroup Hx is called variously the stabilizer subgroup,isotropy subgroup or little group of x. For example, the group SO(3) acts on S2 ⊂ E3

in the standard way and an SO(2) ≡ U(1) subgroup fixes the north pole and southpoles. Another SO(2) subgroup in fact a (different) SO(2) subgroup fixes every pair ofantipodal points. These subgroups are all related by conjugation. Thus if y = φg(x) sothat x = φg−1(y) then

φg φh φg−1(y) = y. (1.16)

That isφghg−1(y) = y , ⇒ Hy = gHxg

−1. (1.17)

Thus, up to conjugation, which corresponds to choosing an origin in X, there is a uniquesubgroup H and one may regard the homogeneous space X as a right coset G/H on whichG acts on the left.

1.5.1 Example:De-Sitter and Anti-de-Sitter Spacetimes

.De-Sitter spacetime, dS4 and anti-De-Sitter spacetime AdS4 are conveniently defined

as quadrics in five dimensional flat spacetimes, with signature (4, 1) and (3, 2) respective-ley:

dS4 : (X1)2 + (X2)2 + (X3)2 + (X4)2 − (X5)2 = 1. (1.18)

AdS4 : (X1)2 + (X2)2 + (X3)2 − (X4)2 − (X5)2 = −1. (1.19)

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They inherit a Lorentzian metric from the ambient flat spactime which is preserved byG = SO(4, 1) or SO(3, 2) which acts transitivley with stablizer H = S0(3, 1) in bothcases. Thus dS4 = SO(3, 1)/S0(3, 1) and dS4 = SO(3, 1)/S0(3, 1).

1.5.2 Example: Complex Projective space

A physical description of complex projective space, CPn is as the space of pure states ofan n+ 1 state quantum system with Hilbert space H ≡ Cn+1. Thus a general state is

ψ〉 =

n+1∑

r=1

Zr|r〉, (1.20)

where |r〉 is a basis for H. Now since |Ψ〉 and λ|ψ〉, with λ ∈ C \ 0 define the samephysical state the space of states is paramtrized by Z1, Z2 . . . Zn+1 but (Z1, Z2 . . . Zn+1) ≡(λZ1, λZ2, . . . , λZn+1), with λ ∈ C \ 0, which is the standard mathematical definition ofCPn which is the simplest non-trivial example of a complex manfold with complex chartsand locally holomorphic transtion functions

Working in an orthonormal basis, we may attempt to reduce the redundancy of thestate vector description by considering normalized states

n+1∑

r=1

|Zr|2 = 1 . (1.21)

This restricts the vectors Ψ〉 to lie on the unit sphereS2n+1 = SO(2n+ 2)/SO(2n+ 1) = U(n + 1)/U(n). However one must still take into

account the redundant, and unphysical, phase of the state vector. The group G = U(n+1acts transitively on CCPn and the stabilizer of each point is H = U(n)× U(1). We maythink of CPn as the set of orbits in S2n+1 of the action of U(1) given by

Zr → eiθZr. (1.22)

Each orbit is a circle S1 and the assigment of each pooint to the orbit through it definesis called the Hopf map

π : S2n+1 → CPn . (1.23)

The structure we have just described provides perhaps the simplest example of a fibrebundle, in fact a circle bundle. We shall treat them in further detail later

The most basic case is the spin 12

system which has n = 1. One has CP1 ≡ S2 ≡SU(2)/U(1) ≡ U(2)/U(1)× U(1). We may introduc two coordinate patches, ζ = Z1/Z2

and ζ = Z2/Z1, one covering the north pole and the other the south pole of S2. On theoverlap ζ = 1

ζ. In fact these are stereographic coordinates.

1.6 Representations

If X is some vector space V and g acts linearly, i.e by endomorphism or linear maps wehave a representation of G by matrices D(g) such that

D(g1)D(g2) = D(g1g2) etc. (1.24)

The representation said to be faithful if the action is effective, i.e. no element of Gother than the identity leaves every vector in V invariant. Another way to say thisis that a representation is a homomorphism D of G into a subgroup of GL(V ) and afaithful representation has no kernel. Given an unfaithful representation we get a faithfulrepresentation by taking the quotient of G by the kernel.

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1.6.1 Reducible and irreducible representations

The representation is said to be irreducible if no vector subspace of V is left-invariant,otherwise it is said to be reducible. The representation is said to be fully reducible if thecomplementary subspace is invariant.

A reducible but not fully reducible representation has

D(g) =

(

· · · 0· · · · · ·

)

. (1.25)

and vectors of the form

(

0. . .

)

are invarint but those of the form

(

· · ·· · ·

)

are not. On the

other hand for a fully reducible representation one has

D(g) =

(

· · · 00 · · ·

)

. (1.26)

and both subspaces are invariant. If V admits a G -invariant definite inner product, Bsay, then any reducible representation is fully reducible.

Proof Use B to project any vector into W , the invariant subspace, and its orthogonalcomplement W⊥ with respect to B Thus for any v ∈ V we have v = v‖ +v⊥. Now G takesv‖ to v‖ ∈W and v⊥ to v⊥ ∈W⊥,

1.6.2 The contragredient representation

If D(g) acts on V then (D−1(g))t acts naturally on V ⋆ the dual space. In components

D(g) : V a → Dab(g)V

b = V a . (1.27)

D−1(g) : ωa → ωc(D−1)c a = ωa (1.28)

(D−1)(g) : (ωt)a → (ωt)a = ((D−1)t)a b(ωt)b . (1.29)

Thusωa V

a = ωaVa . (1.30)

The reason we need to transpose and invert is that otherwise the matrix multiplicationwould be in reverse order. We would have an anti-homomorphism og G into GL(V ) or inour previous language a right action of G on V .

1.6.3 Equivalent Representations

Two representations D and D acting on vector spaces V and W are said to be equivalentif there exist an invertible linear map B : V → B and D = B−1DB.

Example Suppose thatW = V ⋆, the dual space of V , andB is a symmetric non-degneratebillinear form or metric invariant under the action of G, then D and the contragredientrepresentation (D−1)t are equivalenet.

Proof Invariance of B under the action of G requires that

DtBD = B. (1.31)

ThusD = B−1(D−1)tB. (1.32)

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In components(D−1)b a = BacD

ceB

eb. (1.33)

Thus raising and lowering indices with the metric B, which need not be positivedefinite, gives equivalent representions on V and V ⋆. The maps B and B−1 are sometimesreferred to as the musical isomorphisms and denoted by and ♯ respectively.

In fact there is no need for B to be symmetric, B = Bt. It often happens that onehave a non-degenerate bi-linear form which is anti-symmetric, B = −Bt. In that case Vis said to be endowed with a symplectic structure, or symplectic for short.

1.7 *Semi-direct products, group extensions and exact sequences*

Given two groups G and H there is an obvious notion of product

G×H : (g1, h1)(g2, h2)→ (g1g2, h1h2) . (1.34)

. Frequently however, one encounters a more sophisticated construction: the semi-directproduct G⋉H . The Poincare group for example, is the semi-direct product of the Lorentzgroup and the translation group, such that the translations form an invariant subgroupbut the the Lorentz-transformstions do not. The construction also arises when one wantsto extend a group G by another group H . It also arises in the theory of fibre bundleswhich we will outline later.

Suppose G acts on H preserving the group structure, i.e. there is a map ρ : G→ H sothat ρg1(h) ρg2(h) = ρg1g2(h) and ρg(h1)ρg(h2) = ρg(h1h2). If, for example, H is abelian,thought of as additively it is a vector space, then then ρg would be a representation of Gon H . In general ρ is homomorphism of G into Aut(H) the automrphism group of H .

We define the product law by

G⋊H : (g1, h1)(g2, h2)→ (g1g2, h1ρg1(h2)) , (1.35)

which is asociative with inverse

(g, h)−1 = (g−1, ρg−1(h−1) . (1.36)

A standard example is the Affine group A(n) = GL(n,R⋉Rn, which acts on n-dimensionalaffine space An ≡ Rn by translations and general linear transformations preserving theusual flat affine connection. A matrix representation is

(

Λab aa

0 1

)

, Λ ∈ GL(n,R) aa ∈ Rn . (1.37)

we have(

Λ a0 1

) (

Λ′ a′

0 1

)

=

(

ΛΛ′ a+ Λa′

0 1

)

. (1.38)

Restricting toO(n) ⊂ GL(n,R) we obtain the Euclidean groupE(n) = O(n)⋉Rn. ThePoincare group is obtained by restriction to the Lorentz group SO(n− 1, 1) ⊂ GL(n,R).

Glearly G may be regarded as a subgroup of G⋉H because of the injection

i : G→ G⋉H , gi−→ (g, e). (1.39)

One may also regard H as a subgroup of G⋉H because

(e, h1)(e, h2) = (e, h1h2) . (1.40)

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Moreover, H is an invariant subgroup of G×H because

(g, h)(e, h′)(g−1, h−1ρg−1(h−1)) = (e, hρg(h′)ρg−1(h−1)) ∈ (e,H) . (1.41)

By contrast, G is, in general (if ρ 6= id ), not an invariant subgroup because

(g, h)(g′, h′)(g−1, h−1ρg−1(h−1)) (1.42)

= (gg′g−1, hρg(h′ρg′(h

−1ρg−1(h−1))) (1.43)

= (gg′g−1, hρg(h′)ρg′(h)ρg′g−1(h−1)) (1.44)

= (gg′g−1, hρg(h′)ρg−1(h−1)) . (1.45)

Thus even if η′ = e, the second term is not necesarily the identity. In summary, we havewhat is called an exact sequence,

1 −→ Gi−→ G⋉H

π−→ H −→, (1.46) ext

that is the where the maps π : (g, h)→ (e, h), i→ (g, e) satisfy the relation that

Image(i) = Kernel(π) , (1.47)

and the arrows at the ends are the obvious maps of identity elements. Note that in thislanguage a map φ : G→ H is an isomorphism if

1 −→ Gφ−→ H −→ 1 . (1.48)

It is merely a homomorphism if

Gφ−→ H −→ 1 . (1.49)

and G is just a subgroup of H if

1 −→ Gφ−→ H . (1.50)

If (1.46) holds, one says that G ⋉ H is an extension of G by H . If H ⊂ Z(G ⋉ H), thecentre, of one speaks of a central extension. Thus Nil or the Heisenberg group is a centralextension of the 2-dimensional abelian group of translations.

1.7.1 The five Lorentz groups

Other examples of semi-direct products arise when one considers the discrete symmetriesparity P and time reversal T . These are two commuting involutions inside O(3, 1) actingon four-dimensional Minkowski spacetime E3,1 as

(x0, xi)P−→ (x0,−xi) , (x0, xi)

T−→ (−x0, xi) . (1.51)

We abuse notaton by calling P , T , PT , and (P, T ) the Z2 and Z2×Z2 groups generatedby ,P, T, PT and P, T respectivley. Only PT is an invariant subgroup of O(3, 1), andindeed belongs to the centre of O(3, 1). We define the following invariant subsroups:

(i)SO0(3, 1) is the connected component of the identity in O(3, 1),

(ii) SO(3, 1) is the special Lorentz group which is the subgroup with unit determinant.

(iii) O ↑ (3, 1) is the othochronous Lorentz-group which preserves time orientation,

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(iv) SO+(3, 1) as the subgroup preserving space-orientation.One has

O(3, 1)/SO0(3, 1) = Z2 × Z2 O(3, 1) = (P, T ) ⋉ SO0(3, 1)O(3, 1)/SO0(3, 1) = Z2 O(3, 1) = PT ⋉ SO(3, 1)O(3, 1)/O ↑ (3, 1) = Z2 O(3, 1) = T ⋉ O ↑ (3, 1)O(3, 1)/O+(3, 1) = Z2 O(3, 1) = P ⋉ O+(3, 1) .

(1.52)

Thus for example

1 −→ (P, T )i−→ O(3, 1)

π−→ SO0(3, 1) −→ 1 . (1.53)

Geometrically, O(3, 1), for example, has four connected components and may be re-garded as a principal bundle over a a conected base B = SO0(3, 1) with fibres G =Z2 × Z2 = (P, t) consisting of 4 disjoint points.

1.8 Geometry of Lie Groups

Lie groups are

(i) Differentiable Manifolds

(ii) groups

(iii) such that group multiplication and inversion are diffeomrphismsThe Lie algebra g of G may be identified (as a vector space) with the tangent space

of the origin or identity element eg = Te(G) . (1.54)

Now if Cg is conjugation, then it fixes the origin and the derivative map Adg = Cg⋆ actslinearly on Te(g) via the what is called the adjoint representation and the contra-gredientrepresentation acts on g⋆ the dual of the Lie algebra via waht is called the co-adjointrepresentation. In general, these representations are not equivalent although given aninvarint metric on G they will be. Now let ea be a basis for g and ea a basis for g⋆ sothat.

〈ea|eb〉 = δab . (1.55)

Since the right action Rg and left action Lg of G on itself are simply transitive we use thepull-forward and push-back maps Rg⋆

, Lg⋆ and R⋆g, L

sgtar to obtain a set of left-invariant

vector fields La and right-invariant vector fields Ra and left–invariant one forms λa andright-invariant one forms ρa .

ThusRg⋆

ea = Ra(g) R⋆gρa = ea , (1.56)

Lg⋆ea = La(g) R⋆gλ

a = ea , (1.57)

and〈λa|Lb〉 = δab 〈ρa|Rb〉 = δab . (1.58)

Thus, for example,

Lg1⋆La(g2) = Lg1⋆Lg2 ⋆ ea = Lg1g2 ⋆ ea = La(g1g2) etc... (1.59)

Thus every Lie Group is equipped with a global frame field (in two ways). We shalluse these frame fields to do calculations.Note that, because left and right actions are

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global diffeomorphisms, the the frames are everywhere linearly independent. A manifoldadmitting such a global frame field is said to be parallelizable. Thus every Lie group isparallelizable. Among spheres, it is known that only S1, S3 and S7 are parallelizable. Thefirst two are groups manifolds, U(1) and SU(2) respectivley but the third is not. In factthe parallelism of S7 has some applications to supergravity theory and is closely relatedto the fact that S7 may be regarded as the unit sphere in the octonions O , just as S1

and S3 may be regarded as the unit sphere in the complex numbers C and quaternions Hrespectively.

1.8.1 Matrix Groups

The basic example of a Lie Group is SU(2) which we may think of as complex valuedmatrices of the form

(

a b−b a

)

|a|2 + |b|2 = 1 . (1.60)

If a = X1 + iX2 and b = X3 + iX4 this defines a unit 3-sphere S3 ⊂ E4.Closely related is SL(2,R) which we may think of as real valued matrices of the form

(

a bc d

)

ad− bc = 1 . (1.61)

If a = X0 +X3, d = X0 −X3, b = X2 +X4, c = X2 −X4 this defines a a quadric in E2,1.In fact we may identify SL(“,R) with three-dimensional Anti-de-Sitter spacetime AdS3.

The group SL(2,R) acting on R2 preserves the skew two by two matrix ǫAB = −ǫBA =(

0 1−1 0

)

. This is because for two by two matrices

detX = XA1X

B2ǫAB = ǫ12 = 1. (1.62)

In fact SL(2,R) is the first in an infinite series of Lie groups called the symplectic groupsSp(2n,R) which preserve a skew-symmetric form ωab = −ωba. Just as the orthogonalgroup SO(n) is the basis of Riemanian geometry involving asymmetric positive definitebi-linear form or metric gab = gba, the symplectic groups is the basis of symplectic geometry,i.e. the geomery of phase space and involves an anti-symmetric tensor two-form ωab =−ωba.

1.9 Infinitesimal Generators of Right and Left Translations

If a group G acts freely on a manifold M on the left say we focus on the action of a one-parameter subgroup g(t) such that g(t)g(t2) = g(t1 + t2) , g(−t) = g−1(t) and g(0) = e.The orbits in M of of g(t) are curves

x(t) = φg(t)(x) . (1.63)

There is at most one such curve through each point p ∈M and it has a tangent vector

K =dx

dt. (1.64)

Thus we get a vector fieldK(x) onM for each one parameter subgroup ofG. A special caseoccurs when M is a Lie group G and the group action φ is by left translations. Each one

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parameter subgroup of G will have an orbit passing through the origin passing through theunit element or origin e ∈ G and an initial tangent vector dg

dt|t=0 which lies in the tangent

space of the origin Te(G) which we identify with the Lie algebra g. If ea is a basis for g ,then get a map, called the (left) exponential map from R×Te(G) ≡ R×g : (t, V )→ ga(t)by moving an amount t along the orbit through e with initial tangent V say.

1.9.1 Example Matrix groups

Near the origin we writeg = 1 + vaMat+O(t2) . (1.65)

where Ma are a set of matrices spanning g. Then

g(t) = exp tV aMa = 1 + tV aMa +t2

2!(V aMa)

2 +t3

3!(V aMa)

3 + . . . . (1.66)

Thus the exponential map corresponds to the exponential of matrices.

1.10 Right invariant Vector Fields generate Left Translations

and vice versa

.By allowing the initial tangent vector V to range over a basis ea of the initial tangent

space Te(g) ≡ g we get family of vector fields Ka on G which are tangent to the curvesg(t) and are called the generators of left-translations.The give a global paralelization ofG. Since left and right actions commute this paralleization is right-invariant. In fact it isclear that this parallelization must coincide with that defined earlier. That is we may setKa = Ra.

It follows that£La

ρb = 0 = £Raλb . (1.67)

1.11 Lie Algebra

We can endow g with a skew symmetric bracket satisfying the Jacobi identity by setting

[ea, eb] = −£RaRb|t=0 = −[Ra, Rb]t=0 . (1.68)

We define the structure constants by

[ea, eb] = Cacbec . (1.69)

Since the brackets of right-invariant vector fields are determined by their values at theorigin we have

[Ra, Rb] = −Ca c bRc . (1.70)

Indeed this relation may also be taken as a definition of the Lie algebra. Note that theJacobi Identity

[ea, eb], ec] + [[ec, ea]eb] + [[eb, ec], ea] = 0 . (1.71)

follows from the Jacobi identity for vector fields

[[X, Y ], Z] + [[Z,X]Y ] + [[Y, Z], X] = 0 . (1.72)

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1.12 Origin of the minus sign

The sign conventions we are adoting are standard in mathematics but perhaps unfamilarto those whos knowledge of Lie algebras comes from some physics textbooks. Moreover itsi illuminating to track them down.In order to understand a sign it is usually suficient toconsider a particular example and so we shall consider the special case of matrix groups.

We move from e to x1 along ga(t) and from x1tox2 along gb(t) and compare withmoving from e to x3 along gb(t) followed by moving from x3 to x4 along ga(t). Now

ga(t)gb(t) = exp tMa exp tMb = 1 + t(Ma +Mb) +t2

2(M2

a +M2b ) + t2MaMb + . . . . (1.73)

so that

x4 − x3 = gagb − gbga = t2(MaMb −MBMb) + · · · = t2[Ma,Mb] + . . . . (1.74)

From the theory of the Lie derivative

x4 − x3 = (£RbRa)t

2 + · · · = [Rb, Ra]T2 + . . . (1.75)

1.13 Brackets of left-invariant vector fields

If one repeats the execrcise for right actions one must the order of multiplication. Thisleads to a sign reversal and one has

[La, Lb] = CacbLc . (1.76)

1.13.1 Example:Right and Left vector fields commute

This is because the former generate left translations and the latter right translations andthese commute. Thus infinitesimally they must commute, i.e.

[Ra, Lb] = 0 . (1.77)

1.14 Maurer-Cartan Forms

Using the formulad(ω(X, Y ) = Xω(Y )− Y ω(X)− ω([X, Y ]) (1.78)

and substituting ρa(Rb) = δab we get

dρa =1

2Cb

acρb ∧ ρc . (1.79)

Similarly

dλa = −1

2Cb

acλb ∧ λc . (1.80)

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1.14.1 Example:Matrix groups

Clearly

g−1dg = λaMa (1.81)

is invariant under g → hg with g constant. Now take one for d

d(g−1) ∧ dg = dλaMa . (1.82)

But for matricesd(g−1) = −g−1dgg−1 . (1.83)

Thus

dλaMa = −λbMb ∧ λcMc = −1

2λb ∧ λc[Mb,Mc] = −1

2Cb

acλb ∧ λcMa . (1.84)

One can similarly check the opposite sign occurs for the right-invariant forms

dgg−1 = ρaMa (1.85)

1.15 Metrics on Lie Groups

Given any symmetric invertible tensor gab = gba ∈ g⋆⊗Sg⋆ we can construct a left-invariantmetric on G by

ds2 = gabλa ⊗ λb , (1.86)

because Gab are constants and £Rbλb = 0. In general, however the metric so constructed

will not be right-invariant. One calls a metric which is, bi-invariant.Since

L⋆hg = g , ∀g ∈ G, (1.87)

it suffices that g be invariant under conjugation

Ch : g → hgh−1 . (1.88)

Now acting at the origin e ∈ G , conjugation rotates the tangent space into itself via theAdjoint representation Adh = Ch⋆ : g→ g. This in turn acts on g⋆ by the contra-gredientor co-adjoint representation. A necessary and sufficent condition for the tensor gab togive rise to a bi-invariant metric is that it be invariant under the co-adjoint action. Onesuch metric is the Killing metric which we will define shortly. We begin by discussing theinfinitesimal version of the Adjoint action, adjoint action adX(Y ) a linear map g→ g (forfixed X ∈ g ) defined by

Y → [X, Y ] . (1.89)

Taking X = ea we haveadea

(Y ) = [ea, Y ] = Ca , (1.90)

where Ca is some matrix or linear map on g with components

(Ca)bc = Ca

bc . (1.91)

The Jocobi identity

[ea, [eb, Y ]]− [eb, [ea, Y ]] = [[ea, eb], Y ] , (1.92)

12

becomesCaCb − CbCa = Ca

cbCc (1.93)

and therefore the matrices Ca provide a matrix representation of the Lie algbra called theadjoint or regular representation.

Multiplying the Jacobi identity by Cd and taking a trace gives

Tr(CdCaCb − CdCbCa) = Cadb , (1.94)

whereCadb = BdcCa

cb , (1.95)

andBdc = Tr(CdCc) = Tr(CcCd) = Bcd (1.96)

is a symmetric bi-linear form on g called the Killing form.Using the cyclic property of the tarce we have

Cabc = −Cbac . (1.97)

But becauseCa

db = −Cb d a (1.98)

we haveCadb = −Cbda, (1.99)

and henceCabcC[abc] . (1.100)

The tensor Cabc defines a 3-form on g and hence by left or right translation a 3-form onG which by use of the Cartan-Maurer formulae may be shown to be closed.

1.15.1 Example: SU(2)

On SU(2), or on SL(2,R), we obtain a multiple of the volume form on S3 or AdS3

respectively.

1.16 Non-degenerate Killing forms

From now on in this subsection , unless otherwise stated, we assume that Bab is non-degenerate, i.e. detBab 6= −. Thus we have an inverse Bab such that BabBbc = δac .Therefore

Cacb = BceCaeb (1.101)

andCa

cc = CceCace = TrCa = 0 . (1.102)

Groups for which TrCa = 0 are said to be unimodular.

13

1.16.1 Necessary and sufficient condition that the Killing form is non-degenerate

This is that G is semi-simple, i.e. it contains no invariant abelian subgroups.We sahllestablish the necessity and leave the sufficiency for the reader.

If G did contain an invariant abelian subgroup H , let eα span h and ei span thecomplement k so that g = h⊕ k. Since H is abelian

Cαµβ = 0 = Cα

iβ. (1.103)

Since H is invaraint[g, h] ⊂ h , ⇐ Ci

jα = 0 . (1.104)

One calculates and finds thatBαβ = 0 = Bαi . (1.105)

ThusBabδ

aα = 0 . (1.106)

which implies that δaα lies in the kernel of B( , ).

1.16.2 Invariance of the Killing metric under the co-adjoint action

let V a ve the components of a vector V ∈ g then under the adjoint action

(δeaV )b = [ea, V ]b = Ca

bcV

c . (1.107)

For example if G = SO(3) g = R3 and

δv = Ω× v , Ω ∈ g . (1.108)

let ωa be the components of a co-vector ω ∈ g⋆. The co-adjoint action must leaveinvariant 〈ω|V 〉 = ωaV

a. Therefore

δeaωb = −ωcCa c b . (1.109)

The induced action on B ∈ g⋆ ⊗S g⋆ is thus

δeaBcb = −Ca e bBce − Ca e cBeb (1.110)

= −Cacb − Cabc = 0 . (1.111)

It follows from our earlier work on representations that if G is semi-simple, the Adjointand the co-Adjoint representation are equivalent since B is an invariant bi-linear formmapping g to g⋆.

It also follows that every semi-simple Lie group is an Einstein manifold with respectto the Killing metric.

If we set λ = 12

in the previous formulae for the ∇ 1

2

connection we get for the Riccitensor

Reb = −1

4Cd

fbCf

de = −1

4Bde . (1.112)

Moreover, it is clear that since ∇ 1

2

preserves the Killing metric and is torsion free, thenit is the Levi-Civita connection of the Killing metric.

The simplest examples are G = SU(2) = S3 and G = SL(2,R) = AdS3 which areclearly Einstein since they are of constant curvature.

Rabcd = −1

8(BacBbd − BadBbc) . (1.113)

14

1.16.3 Signature of the Killing metric

The Killing form B is, in general, indefinite having one sign for compact directions andthe opposite for non-compact directions. By a long standing mathematical convention,the sign is negative definite for compact directions and positive definite for non-compactdirections. Thus in the Yang Mills Lagangian with g-valued curvatures F 1

µν ,

−1

2gabF

aµνF

bµν (1.114)

gauge invaraince requires that the metric gab be invariant under the co-adjoint actionand if gab is minus the Killing metric, then the group must be compact in order to havepositive energy. In supergravity theories,scalars arise, on which which alllow gab is allowedto depend. That is gab becomes spacetime time-dependent. In that case it is pssoible toarrage things to that the gauge group is non-compact while retaining positive energy.

We now illustrate the theory with some examples.

1.16.4 Example SO(3) and SU(2)

These are clearly compact groups since RP3 ≡ S3/Z2 is a compact manifold. In ourconventions Ci

jk = −ǫijk. Thus Bij = ǫirsǫjrs = −2δij . It is clear that since SO(3) acts

in the usual way on its Lie algebra R3, any bi-invariant metric must be a multiple ofδij. For example think of its double cover SU(2) as 2 × 2 unimodular unitary matricesU = (U−1)t, detU = 1. We may set

U =

(

a b−b a

)

, |a|2 + |b|2 = 1 . (1.115)

The left action is U → LU , L ∈ SU(2) The right action is U → UR, R ∈ SU(2).The metric

det dU = |da|2 + |db|2 (1.116)

is invariant under dU → LdUR, and hence bi-invariant.

1.16.5 Matrix Groups

In general, for a matrix group, the Killing metric is proportional to

Trg−1Dgg−1dg = λa ⊗ λbTr(MaMb). (1.117)

For orthogonal groups the infinitesial generators are skew symmetric and Ma = −M ta the

Killing form is negative definite and these groups are compact.

1.16.6 Example:SL(2, R)

As before

g =

(

X0 +X1 X2 +X4

X2 −X4 X0 −X1

)

, (X0)2 + (X4)2 − (X1)2 + (X2) = 1. . (1.118)

A calculation analogous to that above gives the signature + + 1. The group hastopology S1 × R2 with the compact direction corresponding rotations in the 1-2 plane.

15

1.16.7 Example: SO(2, 1) = SL(2,R/Z2

If SO(2, 1) acts on E2,1 then

e3 = x∂y − y∂x generates rotations in x− y plane, (1.119)

e1 = t∂x + x∂t generates boosts in t− x plane, (1.120)

e2 = t∂y + y∂t generates boosts in t− y plane . (1.121)

Thus[e1, e2] = e3 , C1

22 = −1 . (1.122)

[e3, e2] = e1 , C31

2 = −1 . (1.123)

[e3, e1] = −e2 , C12

1 = +1 . (1.124)

Therefore the Killing form is diagonal with B33 = −2 , B11 = B22 = +2. This compatiblewith the fact that the group SO(2) is compact and the group SO(1, 1) is non-compactsince for example as matrices they are

(

cosψ − sinψsinψ cosψ

)

, 0 ≤ ψ < ψ, (1.125)

but(

cosh φ sinhφsinhφ cosh φ

)

, ∞ < φ <∞ . (1.126)

1.16.8 Example: Nil or the Heisenberg Group

This is upper-triangular matrices

g =

1 x z0 1 y0 0 1

= 1 + xX + yY + zZ (1.127)

with

X =

0 1 00 0 00 0 0

(1.128)

Y =

0 0 00 0 10 0 0

(1.129)

Z =

0 0 10 0 00 0 0

(1.130)

(1.131)

and only non-vanishing commutator being

[X, Y ] = Z . (1.132)

16

Thus Z is in the centre Z(g) of the algebra, that is it commutes will all elements in g.The Killing form B is easily seen to vanish. Now

g−1 =

1 −x −z + xy0 1 −y0 0 1

, (1.133)

Thusg−1dg = Xdx+ Y dy = Z(dz − xdy). (1.134)

One finds that

λx = dx , Lx = ∂x , (1.135)

λy = dy , Ly = ∂y + x∂z (1.136)

λz = dz − xdy , Lz = ∂z . (1.137)

The non-trivial Cartan-Maurer relations and Lie brackets are

dλz = −λx ∧ λy , [Lx, Ly] = Lz . (1.138)

Using the right-invariant basis one gets

ρx = dx , Rx = ∂x + ∂z , (1.139)

ρy = dy , Ry = ∂y (1.140)

ρz = dz − ydx , Rz = ∂z . (1.141)

The non-trivial Cartan-Maurer relations and Lie brackets are now

dρz = +ρx ∧ ρy , [Rx, Ry] = −Rz . (1.142)

An example of a left-invariant metric on Nil is

(dz − xdy)2 + dx2 + dy2 . (1.143)

The Lie algebra nil is of course the same as Heisenberg algebra [x, p] = i h2π

id.

1.16.9 Example: Two-dimensional Euclidean Group E(2)

This provides a managable example which illustrates more complicated cases such as thefour-dimensional Poincare group. A convenient matrix representation is

g =

cosψ − sinψ xsinψ cosψ y

0 0 1

. (1.144)

With 0 ≤ ψ < 2π, −∞ < x∞, −∞ < y < ∞. The group is not semi-simple, it is infact a semi-direct product of R2 ⊲⊳ SO(2). The translations form an invariant abeliansub-group The Killing form is degenerate, vansihing on the translations.

The matrix can be thought of as acting on the column vector

X1

X2

1

. (1.145)

17

The subspace

X1

X2

0

(1.146)

is invariant but the complementary subspace

00X3

(1.147)

is not. The representation, which exhibits E(2) as a subgroup of SL(3,R, is thus reduciblebut not fully reducible. Geometrically, this is a projective construction.. Euclidean spaceE2 is identified with those directions through the origin of R3 which intersect the planeX3 = 1. The full projective group of two dimensions is SL(2,R which takes straight linesto straight lines.

1.17 Rigid bodies as geodesic motion with respect to a left-

invariant metric on SO(3).

Consider a rigid body, a lamina constrained to slide on a plane E2. Every configurationmay be obtained by acting with E(2) on some standard configuration in a unique fashion,.Thus the configuration spaceQmay be regarded asE(2). A dynamical motion correspondsto a curve γ(t) in E(2). If there are no frictional forces the motion is free and will bedesribed by geodesic motion with repsct to the metric given by the kinetic energy

T = gµν xµxν , (1.148)

where xµ are local coordinates on Q.The symmetries of the situtation dictate that themetric is left-invariant and this gives rise to conservation of momentum and angularmomentum.

This example obviously generalizes to a rigid body moving in in vacuo in E3 with E(2)replaced by E(3). The model would still apply if the rigid body moves in an incompressiblefrictionless fluid, an as such has been studied by fluid dynamicists in the nineteenthcentury. Note that in this case, the kinetic energy gets renormalized compared with itsvalue in vaucum, becaues one must take into account the kinetic energy of the fluid. Thereis also an obvious generalization to a rigid body moving in certain curved spaces such ashyperbolic spaces Hn = S(n, 1)/SO(n) or a spheres Sn = SO(n+1)/SO(n) and this ideaformed the basis of Helmholtz’s physical characterization of non-Euclidean geometry interms of the axiom of the free-mobility of a rigid body which he thought of as some sortof measuring rod.

The simplest case to consider is when the mody is in moving in E3 but one point inthe body is fixed. The configuration space Q now reduces to SO(3). Each point in thebody has coordinates r(t) in an inertial or space-fixed coordinate system with origin thefixed point , and where r(t) = O(t)x, where x co-moving coordinates fixed in the bodywith respect to some body-fixed frame and O(t) is a curve in SO(3) so that.

O−1(t) = Ot(t) , (1.149)

that is,O(t) is an orthogonal matrix for each time t. Now

18

Left actions correspond to rotations of with respect to the body fixed frame

r→ Lr , i.e. O→ LO , L ∈ SO(3) (1.150)

Left actions correspond to rotations of with respect to the body fixed frame

x→ R−1x , i.e. O → OR , R ∈ SO(3) . (1.151)

The kinetic enery should be invariant under rotatations about inertial axes and thisinvriance gives rise to conserved angular momenta. Thus we anticipate that the kineticenergy T is left-invariant but not in general right-invariant. If ρ(x) is the mass density,T is given by

T =

∫

Body

d3 x1

2ρ(x) r2 (1.152)

=

∫

Body

d3 x1

2ρ(x) rt r (1.153)

=

∫

Body

d3 x1

2ρ(x) (Ox)t Ox (1.154)

=

∫

Body

d3 x1

2ρ(x)xt Ot O x (1.155)

=

∫

Body

d3 x1

2ρ(x)x (O−1O)t(O−1O)x (1.156)

=

∫

Body

d3 x1

2ρ(x)xt ωt ω x (1.157)

(1.158)

whereωij = (O−1O)ij . (1.159)

In three dimensions we may dualize

ωik = (O−1O)ikǫijkωj (1.160)

where ωi are the instantantaneous angular velocities with reepect to a body fixed frame andfind

T =1

2ωiIijωj (1.161)

where

Iij ∈ so(3)⋆ ⊗S so(3)⋆ =

∫

Body

d3ρ(x) (δijx2 − xixj) = Iji (1.162)

is the (time-independent) inertia quadric of the body which serves as a left-invariantmetric on SO(3). Under right actions, i.e. under rotations of the body with respect to aspace-fixed frame

Iij → RikIklRlj , (1.163)

where Rik are the components of an orthogonal matrix. We may use this freedom todiagonalize the inertia quadric

T =1

2Ix(ω

x)2 +1

2Iy(ω

y)2 +1

2Iz(ω

z)2 (1.164)

where Ix, Iy, Iz are called principal moments of inertia.

19

1.17.1 Euler angles

A geometrical argument shows that any triad may be taken into any other triad by anelement of SO(3) of the form may be written as the product of a rotation through anangle ψ about the third axis followed by a rotation through an angle θ about the newfirst axis followed by a rotation through an angle φ about the new third axis

O(ψ, θ, φ) = Rz(φ)Rx(θ)Rz(ψ) (1.165)

=

cosφ − sin φ 0sinφ cosφ 0

0 0 1

1 0 00 cos θ − sin θ0 sin θ cos θ

cosψ − sinψ 0sinψ cosψ 0

0 0 1

,(1.166)

with 0 ≤ φ < 2π, 0 ≤ θ < π, 0 ≤ ψ < 2π. Thus

O−1(ψ, θ, φ) = O(−φ,−θ,−ψ) . (1.167)

A covering element of SU(2) is

S = expφτz2i

expθτx2i

expψτz2i

, (1.168)

so that

S−1dS =λxτx2i

+λyτy2i

+λzτz2i

, (1.169)

with

λx = sin θ sinψdφ+ cosψδθ (1.170)

λy = sin θ cosψdφ− sinψdθ (1.171)

λz = dψ + cos θdφ , (1.172)

and

Lx = cosψ ∂θ − sinψ (cot θ ∂ψ −1

sin θ∂φ) (1.173)

Ly = − sinψ ∂θ − cosψ (cot θ ∂ψ −1

sin θ∂φ) (1.174)

Lz = ∂ψ . (1.175)

The right-invariant basis satisfies

dSS−1 =ρxτx2i

+ρyτy2i

+ρzτz2i

, (1.176)

and may be obtained form the observation that inversion (φ, θ, φ)→ (−φ,−θ,−ψ) takesS−1dS → −dSS−1 and thus

with

−ρx = − sin θ sinφdψ − cos φδθ (1.177)

−ρy = sin θ cosφdφ+ sin φdθ (1.178)

−ρz = −dφ− cos θdψ , (1.179)

20

and

−Rx = − cosφ ∂θ + sinφ (cot θ ∂φ −1

sin θ∂ψ) (1.180)

−Ry = sinφ ∂θ − cosφ (cot θ ∂φ −1

sin θ∂ψ) (1.181)

−Rz = −∂φ . (1.182)

We have

dλx = −λy ∧ λz , [Lx Ly] = Lz etc (1.183)

dρx = ρy ∧ ρz , [Rx, Ry] = −Rz etc . (1.184)

∂∂φ

generates left actions of SO(2) (rotations about the 3rd space-fixed axis)

∂∂ψ

generates right actions of SO(2) ( rotations about the 3rd body-fixed axis)The kinetic energy metric may be written as

ds2 = Ix(sin θ sinψdφ+cosψdθ)2+Iy(sin θ cosψdφ−sinψdθ)2+Iz( ψ+cos θdφ)2 . (1.185)

∂∂φ

is always a Killing vector corresponding to conservation of angular momentum aboutthe third body fixed axis. In general, the full set of Killing vector fields is Rx, Ry, Rz

£Rxds2 = 0 etc . (1.186)

∂∂ψ

is only a Killing vector if Ix = Iy = I in which case

ds2 = Iz(dψ + cos θ)2 + I(dθ2 + sin θ2dφ2) .( (1.187)

In mechanics this is Lagrange’s symmetric top. In geometry S3 = SU(2) equippedwith such a metric is such is sometimes called a Berger-Sphere. This requires our passingto the double cover by allowing the angle ψ, which is an angle along the S1 fibres of theHopf fibration, to run between 0 and π. In general,if we take 0 ≤ 4π

kwe could take a

quotient to get the lens spaces L(k, 1) ≡ S3/Ck, where Ck is the cyclic group of order k.We shall later learn that these are the possible circle bundles over the 2-sphere S2, withk the Chern number. They correspond to all possible Dirac mononopoles on S2 and themetric provides their Kaluza-Klein description of these monopoles.

1.17.2 SL(2,R) and the Goedel Universe

Using the ‘cheap and dirty‘trick of replacing θ → i, χreal results in

λx → iλx = λ1, non− compact (1.188)

λy → iλy = λ2, non− compact (1.189)

λz → iλz = λ0 compact . (1.190)

Up to an overall scale, the family of left-invariant Berger metrics on SU(2) with anadditional SO(2) right action are given by

ds2 = λ(dψ + cos θdφ)2 + (dθ2 + sin2 θdφ2) . (1.191)

21

become a family of metrics on SL(2,R)

ds2 = λ(dψ + coshχφ)2 − (dχ2 + sinh2 χdφ2 . (1.192)

= λ(dt+ 2 sinh2(χ

2)dφ)2 − (dχ2 + sinh2 χdφ2 , (1.193)

with t = ψ + φ. If λ = 1 we get the standard metric on Ad3. If λ > 1 we get a family ofLorentzian metrics . For a certain value of λ we get a metric, which if multiplied by a line,gives Goedel’s homogeoneous solution of the Einstein equations containing rotating dust.If we chose −∞ < t <∞ the manifold is simply connected and in fact homeomorphic toR3. It may be thought as an R bundle over the two-dimensional hyperbolic plane H2. Toanticipate a later section, the bundle is trivial and t provides a global section. However, ifλ > 1 , t is a time function, i.e. a function on a Lorentzian mnfold which increases alongevery timelike curve. We have

gφφ = − sinh2 χ+ 4λ sinh2(χ

2), . (1.194)

Now φ must be periodic period 2π in order to avoid a conical singularity at χ = 0.Thus the curves t = constant, θ = constant are the orbits of an SO(2) action and hencecircles. But if λ > 1 they are timelike, gφφ > 0, for sufficiently large χ. Thus the metricadmits closed timelike curves or CTC’s which is incompatible with the existence of a timefunction.

1.17.3 Example: Minkowski Spacetime and Hermitian Matrices

We can set up a one-one correspondence between E3,1 = (t, x, y, z) and two by two her-mitian matrices x = x† by

x =

(

t+ z x+ iyx− iy t− z

)

, ⇒ det x = t2 − x2 − y2 − z2 . (1.195)

There is an action of S ∈ SL(2,C) preserving hermiticity and the determinant

x→ SxS† , ⇒ det x→ det x . (1.196)

Thus we get a homomomorphism SL(2,C → SO0(3, 1), the identity component of theLorentz group. The kernel, i.e. the pre-image of 1 ∈ SL(2,C) is ±11 = Z2 and thus

SO0(3, 1) = SL(2,C)/Z2 . (1.197)

Evidently SL(2,C acts faithfully on C2 preserving the simplectic form

(

0 −11 0

)

. Ele-

ments of of C2 are Weyl spinors. One has Spin(3, 1) = SL(2,C). Specialization to SU(2)yields Spin(3) = SU(2), i.e. SO(3) = SU(2)/Z2.

Working over the reals

x =

(

t+ z x+ yx− y t− z

)

, ⇒ det x = t2 − x2 + y2 − z2 . (1.198)

There is an action of S × S ′ ∈ SL(2,R× SL(2,R) preserving the determinant

x→ SxS ′ , ⇒ det x→ det x . (1.199)

22

Thus we get a homomomorphism SL(2,R)× SL(2,R)→ SO0(2, 2). The kernel, i.e. thepre-image of 1 ∈ SL(2,R)× SL(2,R) is ±(1, 1) = Z2 and thus

SO0(2, 2) = SL(2,R)× SL(2,R)/Z2 . (1.200)

We have Spin(2, 2) = SL(2R)×SL(2,R and setting y = 0 and S = S ′ yields Spin(2, 1) =SL(2,R), SO0(2, 1) = SL(2,R)/Z2. Elements of R2 or R2 ⊕ R2 are called Majoranaspinors for Spin(2, 1) or Spin(2, 2) respectivley. Elements of 0⊕ R2 or R2 ⊕ 0 are calledMajorana -Weyl spinors for Spin(2, 2).

1.17.4 Example: SO(4) and quaternions

Ifi =

τ1√−1

j =τ2√−1

, k =τ1√−1

, (1.201)

then we have the algebra

ij + ji = jk + kj = ki+ ik = 0 , (1.202)

ij − k = jk − i = ki− j = 0 , (1.203)

i2 + 1 = j2 + 1 = k2 + 1 = 0 . (1.204)

It may be verified that the quaternion algebra H coincides with the real Clifford algebra,Cliff(0, 2) of E0,2 generated by gamma matrices γ1, γ2 such that γ2

1 = γ22 = −1. One sets

γ3 = γ1γ2 = −γ2γ1 and identifies (i, j, k) with (γ1, γ2, γ3).A general quaternion q ∈ H and its conjugate q are given by

q = τ + xi+ yj + zk, , q = τ − xi− yj − zk . (1.205)

Thusqq = qq = |q|2 = τ 2 + x2 + y2 + z2 . (1.206)

We may thus identify H with four-dimensional Euclidean space E4 and if

q → lqr (1.207)

with |r| = |l| = 1, we preserve the metric. We may identify unit quaternions with SU(2) =Sp(1) and similar reasoning to that given earlier shows that SO(4) = SU(2)×SU(2))/Z2,Spin(4) = Sp(1)×Sp(1), where Sp(n) is the group of n times n quaternion valued matriceswhich are unitary with respect to the quaternion conjugate.

1.17.5 Example: Hopf-fibration and Toroidal Coordinates

The Hopf fibration of S3 is the most basic example of a non-trivial fibre bundle. It isuseful therefore to have a good intuitive understanding of its properties. One way toacquire this is by stereogrphic projection into ordinary flat 3-space .

We may think of S3 as (Z1, Z2) ∈ C2 such that |Z1|2 + |Z2|2 = 1. We set

Z1 = X1 + iX2 = eiα tanh σ , X2 = X4 + iX3 = eiβ1

cosh σ. (1.208)

The surfaces σ = constant or |Z1| =√

1− |Z2|2 = constant are called Clifford tori andα and β are coordinates on each torus. The Hopf fibration sends (α, β)→ (α + c, β + c)

23

and the anti-Hopf fibration sends (α, β) → (α + c, β − c), c ∈ S1 and the fibres spiralaround the torus.There are two degenerate tori, σ = 0, |Z1| = 0, and σ = ∞, |Z2| = 0,which collapse to circles. The induced round metric on S3 is

dσ2

cosh2 σ+ tanh2 σdα2 +

1

cosh2 σdβ2 . (1.209)

This looks more symmetrical if one sets sin = tanhσ,0 ≤ ≤ π2

ds2 = d2 + sin2 ωdα2 + cos2dβ2 . (1.210)

The torus = π4

is square and a minimal surface. Stereographic projection may beachieved by setting

Z1 = sinχ sin θeiφ , Z2 = cosχ + i sinχ cos θeiφ , (1.211)

where χ, θ, φ are polar coordinates on S3 in which the metric is

ds2 = dχ2 + sin2 χ(dθ2 + sin2 dφ2) (1.212)

=4

(1 + r2)2(dr2 + r2(dθ2 + sin2 dφ2)) (1.213)

with r = tan χ

2.

If (x, y, z) = (r sin θ cosφ, r sin θ sin φ, cos θ) are Cartesian coordiantes for E3, on findsthat

x+ iy = eiαsinh σ

cosh σ cosβ, z =

sin β

cosσ + cos β, (1.214)

and

(√

x2 + y2 − cothσ)2 + z2 =1

sinh2 σ. (1.215)

Thus we obtain the standard orthogonal coordinate system on E3 consisting of tori ofrevolution σ = constant obtained by rotating about the z-axis circles parameterized bythe coordinate β lyinging in the planes φ = constant. Shifting α corresponds to rotatingabout the z-axis. The coordinate β. The two degenerate tori σ = 0 and σ =∞ correspondto the z-axis and to a circle lying in the xy plane respectively. The flat metric turns outto be

ds2flat =

1

(cosh σ + cosβ)2(dσ2 + dβ2 + sinh2 dα2) . (1.216)

The reader should verify that the linking number of any disjoint pairs of Hopf fibresis 1.

1.17.6 Example: Kaluza-Klein Theory

Consider the time-independent Schrodinger equation on on the Heisenberg group withleft-invarinat metric

ds2 = (dz − zdy)2 + dx2 + dy2 . (1.217)

One finds that−∇2φ = Eφ⇒ φxx + φzz + (∂+x∂zz)

2φ = −Eφ . (1.218)

If one separates variables φ = Ψ(x, y)eiez one gets

Ψxx − e2Ψ + (∂y + iex)2Ψ = −EΨ . (1.219)

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This is of the form(∂x − ieAx)2Ψ + (∂y − ieAy)2Ψ = −EΨ (1.220)

withA = −xdy ⇒ F = −dx ∧ dy . (1.221)

This is the Schrodinger equation for the Landau problem, i.e. that of a particle of chargee moving in the plane in a uniform magnetic field. If the coordinate z is periodic,0 ≤ z ≤2πL then the electric charge e is quantized eL ∈ Z.

A more general case works as follows. Let 2 WE have ametric on M

ds2 = (z) + Aµdxµ)2 + gµνdx

µdxν , (1.222)

with Killing vector field ∂∂z

generating an S1 = SO(2) isometry groups. The point particleLagrangian on M

(z + Aµxµ)2 = gµν x

µxν . (1.223)

The conserved charge isz + Aµx

µ = c . (1.224)

The equation of motion for xm turns out to be

d2xσ

dt2+

µσν

xµxν = cgστFτµxµ (1.225)

with Fµν = ∂µAν − ∂νAµ. This is a charged geodesic motion in the quotient or basemanifold B = M/SO(2).

As an example consider geodesics of the Berger spheres

L = (ψ + cos θφ)2 + sin θ2φ+ θ2 . (1.226)

We have A = sin θdφ ⇒ F = dA = − sin θdθ ∧ dφ. This is a magnetic monopole onthe B = S2 = SU(2)/SO(2). Electric charge quantization becomes angular momentumconservation in the extra dimension, a view point with a curiously echo of the ideas of LordKelvin and other nineteenth century physicists based on the Corioli force, that magneticforces were essentially gyroscpic in origin..

The reader should verify that the orbits are all little circles on S2 with a size fixed bythe magnitude of the charge, or equivalently the radius. One may repeat the problem onthe Hyperbolic plane H2 = PSL(2,R = SO0(2, 1). If this is mapped into the unit discin the complex plane, one finds that all orbits are circles. For weak magnetic field thecircles intersect the circle at infinity. For large enough magnetic field, the orbits are closedand lie inside the unit disc. If we pass to a compact quotient Σ = H2/Γ, Γ a suitablediscrete subgroup of PSL(2,R, then it is known that the geodesic motion on an energyshell is ergodic. The charged particle motion remains ergodic as long as the orbits on thecovering space are closed circles.

2Experts will realise that, for simplicity, we have set the ‘gravi-scalar’to a constant value

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