Marathe K. Topics in Physical Mathematics

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98 3 Manifolds

H is a Lie subgroup and the quotient G/H with the natural transitive action

of G is a G-homogeneous space.

Example 3.10 The rotation group SO(n +1) of R

n+1

acts on the sphere

transitively. The isotropy group at the point (1, 0,...,0) ∈ R

n+1

may be

identiﬁed with SO(n). Thus, the sphere S

is a homogeneous space of the

group SO(n +1), i.e.,

= SO(n +1)/SO(n).

The conformal group SO(n, 1), deﬁned in Example 3.12, acts transitively

on the open unit ball B

⊂ R

n+1

with isotropy group SO(n) at the origin.

Thus B

is a homogeneous space of the conformal group, called the Poincar´e

model of hyperbolic space, i.e.

= SO(n, 1)/SO(n).

For n =5this construction occurs in the study of the moduli space of BPST

instantons (see Section 9.2).

A G-action on M is said to be free if gx = x for some x ∈ M implies

g = e, i.e., if H

= {e}, ∀x ∈ M.AG-action on M is said to be eﬀective if

gx = x, ∀x ∈ M implies g = e.

A vector ﬁeld X ∈X(M)issaidtobeinvariant under the action L of

G on M (or G-invariant)if

◦ X = X ◦L

, ∀g ∈ G;

i.e., the following diagram commutes:

TM TM

Equivalently, X is G-invariant if

)

∗

X = X, ∀g ∈ G.

G-invariant tensor ﬁelds and G-invariant diﬀerential forms can be

deﬁned similarly. A right G-action on M may be deﬁned similarly; then,

with obvious changes, one has the related notions of orbit, transitive action,

etc.

We now consider an important example of a natural action of G when the

manifold is G itself and the action is by left multiplication. A vector ﬁeld

X ∈X(G)issaidtobeleft invariant if it is invariant under this action

3.6 Lie Groups 99

by left multiplication. The set of all left invariant vector ﬁelds is a ﬁnite-

dimensional Lie subalgebra of the Lie algebra X(G) and is called the Lie

algebra of the group G and is denoted by LG or g. The tangent space T

to G at the identity e is isomorphic, as a vector space, to g. This isomorphism

is used to make T

G into a Lie algebra isomorphic to g. Precisely, for C ∈ T

we deﬁne the left invariant vector ﬁeld X

on G by

(g):=TL

(C), ∀g ∈ G,

where L

: G → G is left multiplication by g.NowforA, B ∈ T

G the Lie

product [A, B] is deﬁned by

[A, B]:=[X

](e) ∈ T

where X

are the left invariant vector ﬁelds corresponding, respectively,

to A and B.LetG, H be Lie groups with Lie algebras g, h respectively. Let

f : G → H be a Lie group homomorphism and let f

∗

: g → h be the map

deﬁned by

∗

):=X

, where

A := T

f(A) ∈ T

i.e., f

∗

) is the left invariant vector ﬁeld on H determined by

A ∈ T

Then one can show that f

∗

is a Lie algebra homomorphism of g into h.

The element A ∈ g generates a global one-parameter group φ

of diﬀeo-

morphisms of G, determined by the global ﬂow φ(t, g)ofA.Thusφ

(e)isthe

integral curve of the left invariant vector ﬁeld A through e.Weobservethat

(e) is the one-parameter subgroup of G generated by A ∈ g.Conversely,a

one-parameter subgroup φ

of G determines a unique element A ∈ g,namely

the tangent to φ

at e ∈ G. These observations play an important role in the

construction of Lie algebras. We deﬁne the map exp : g → G by

exp : A → φ

(e).

This map is a homeomorphism on some neighborhood of 0 ∈ g and can be

used to deﬁne a special coordinate chart on G called the normal coordinate

chart.Themapexp is called the exponential map and coincides with the

usual exponential function for matrix groups and algebras. One can show

that t → exp(tA) is the integral curve of A through e ∈ G, i.e.,

exp(tA)=φ

(e).

The adjoint action Ad of G on itself is deﬁned by the map

Ad : h → Ad(h)ofG → Aut G,

where Ad(h):G → G is deﬁned by

Ad(h)g = hgh

−1

100 3 Manifolds

This action induces an action ad of G on g, which is a representation of G,

called the adjoint representation of G on g. We observe that with the

identiﬁcation of T

G with g,

ad g = T (Ad(g))|

The set GL(g) of linear invertible transformations of g is a Lie group with

Lie algebra: the algebra gl(g) of linear transformations of g with Lie product

given by

[M,N]=M ◦ N − N ◦M, ∀M, N ∈ gl(g).

Thus the restriction of (ad)

∗

to g, also denoted by ad

∗

,isamapofg into

gl(g). If V is a Lie algebra, let us denote by ad the map

ad : V → gl(V )

deﬁned by

ad(X)(Y )=[X, Y ], ∀X,Y ∈ V. (3.19)

The map ad is a Lie algebra homomorphism called the adjoint represen-

tation of V .InthecaseofV = g one can show that ad

∗

coincides with ad

deﬁned in equation (3.19). Therefore, we use ad to also denote ad

∗

The contragredient of the representation ad of G on g, called the coadjoint

representation of G on g

∗

, is denoted by ad

∗

.Thus,∀g ∈ G,

∗

(g)=(ad(g

−1

))

: g

∗

→ g

∗

is the transpose of the linear map ad(g

−1

). It plays an important role in the

theory of representations of nilpotent and solvable groups. The study of this

representation is the starting point of the Kostant–Kirillov–Souriau theory

of geometric quantization (see Abraham and Marsden [1]and[271,272]and

references therein). We now give a construction of the Lie algebras of the Lie

groups GL(n, R)andGL(n, C)deﬁnedinExample3.5 as well as those of

SL(n, R)andSL(n, C)deﬁnedinExample3.9.

Example 3.11 We ﬁrst observe that, when the diﬀerential manifold M is

an open subset of a Banach space F ,thenTM = M ×F .TheLiegroupG ≡

GL(n, R) is an open subset of the n

-dimensional Banach space M (n, R).

Then TG = GL(n, R) × M (n, R) and thus the Lie algebra of GL(n, R) can

be identiﬁed with M (n, R) as a vector space. If A ∈ M (n, R), then the left

invariant vector ﬁeld associated to A is

: GL(n, R) → GL(n, R) × M(n, R)

deﬁned by

(g)=(g, gA).

3.6 Lie Groups 101

From this it follows that the Lie algebra structure of M (n, R) is the one

given by the commutator. Analogously, the Lie algebra of GL(n, C) is the Lie

algebra M (n, C) with the Lie algebra structure given by the commutator. We

note that the map exp deﬁned above becomes in this case the usual exponential

map deﬁned by

exp(A)=e

= I +

+ ···.

In view of the fact that det(e

)=e

tr(A)

, we conclude that

sl(n, R):={A ∈ M (n, R) | tr(A)=0}

is the Lie algebra of SL(n, R). Similarly,

sl(n, C):={A ∈ M(n, C) | tr(A)=0}

is the Lie algebra of SL(n, C).

We now give several examples of Lie groups and their Lie algebras, which

appear in many applications.

Example 3.12 Let g

be the canonical pseudo-metric on R

of signature

n − k, i.e., g

is the pseudo-metric on R

whose matrix representation, also

denoted by g

,is



0 −I

n−k



where I

denotes the i ×i unit matrix, i = k, n −k,andthe0 denote suitable

zero matrices. Let O(k, n−k) be the group of linear transformations A of R

such that g

(Ax, Ay)=g

(x, y), ∀x, y ∈ R

. The group O(k, n − k) can be

identiﬁed with the group of n × n matrices A such that

A = g

From this equation it follows that det A = ±1, ∀A ∈ O(k, n − k).Wedenote

by SO(k, n−k) the subgroup of O(k, n−k) of matrices A such that det A =1.

We write O(n), SO(n) in place, respectively, of O(n, 0), SO(n, 0). The group

O(n) (resp., SO(n)) is the orthogonal group (resp., special orthogonal group)

in n dimensions deﬁned in Example 3.9.Letf : GL(n, R) → S(n, R) be the

map deﬁned by

f(A)=A

The map f is smooth and then f

−1

({g

}) is closed. Thus, O(k, n − k)=

−1

({g

}) is a closed subgroup of GL(n, R) and thus is a Lie subgroup of

GL(n, R). For any A ∈ GL(n, R), the linear map Df(A) on M(n, R) is

given by

Df(A)B = B

A + A

B, ∀B ∈ M(n, R).

Thus,

102 3 Manifolds

Ker Df(I

)={B ∈ M (n, R) | B

= −g

B}.

This implies that the rank of f at I

is maximum (it is equal to n(n+1)/2)and

that f is a submersion at I

. By Theorem 3.1, the Lie algebra of O(k, n −k),

denoted by o(k, n − k), can be identiﬁed with Ker T

f and thus is given by

o(k, n − k)={B ∈ M(n, R) | B

= −g

B}.

This also implies that the dimension of O(k, n−k) is n(n−1)/2.Inparticular,

with g

= I

, the Lie algebra o(n) of O(n) is given by

o(n)={B ∈ M (n, R) | B

= −B},

the Lie algebra of the antisymmetric n × n matrices. It is easy to see that

SO(k, n −k) is a Lie subgroup of O(k,n −k) of dimension n(n −1)/2, given

by the connected component of the identity element. Its Lie algebra denoted

by so(k, n − k) is given by

so(k, n − k)={B ∈ o(k, n − k) | tr(B)=0}.

The groups O(n), SO(n) are compact.

Example 3.13 Let us denote by g

the canonical non-degenerate Hermitian

sesquilinear form on C

of signature n −k, i.e. the sesquilinear form on C

given by

(x, y)=x

+ ···+ x

− x

k+1

−···−x

∀x, y ∈ C

. Then the matrix representation of g

is the same as for g

the previous example. Let U (k, n − k) be the group of linear transformations

A of C

such that g

(Ax, Ay)=g

(x, y), ∀x, y ∈ C

. The group U(k, n −k)

identiﬁes with the group of n × n complex matrices A such that

†

A = g

where A

†

denotes the conjugate transpose of A. From this equation it follows

that det A = ±1, ∀A ∈ U(k, n − k).WedenotebySU(k, n −k) the subgroup

of U(k, n − k) of matrices A such that det A =1.WewriteU (n), SU(n)

in place of U(n, 0), SU(n, 0), respectively. The group U(n) (resp., SU(n))is

the unitary group (resp., special unitary group) in n dimensions deﬁned in

Example 3.9.Letf : GL(n, C) → S(n, C) be the map deﬁned by

f(A)=A

†

The map f is smooth and hence f

−1

({g

}) is closed. Thus, U (k, n − k)=

−1

({g

}) is a closed subgroup of GL(n, C) and thus a Lie subgroup of

GL(n, C). Proceeding as in the previous example, we ﬁnd that the Lie al-

gebra of U(k, n − k), denoted by u(k, n − k), is given by

3.6 Lie Groups 103

u(k, n − k)={B ∈ M(n, C) | B

†

= −g

B}.

This also implies that the dimension of U(k, n −k) is n

. In particular, with

= I

, the Lie algebra u(n) of U(n) is given by

u(n)={B ∈ M (n, C) | B

†

= −B},

the Lie algebra of the anti-Hermitian n ×n matrices. One easily realizes that

SU(k, n −k) is a Lie subgroup of U (k, n −k) of dimension n

− 1, with Lie

algebra su(n, n − k) given by

su(k, n − k)={B ∈ u(k, n − k) | tr(B)=0}.

The groups U(n), SU(n) are compact.

Example 3.14 Let ω be the canonical symplectic structure on R

whose

matrix representation, also denoted by ω,is

ω =



0 I

−I 0



where I (resp., 0) denotes the n × n unit (resp., zero) matrix. Let SP(n, R)

be the group of linear transformations A of R

, such that ω(Ax, Ay)=

ω(x, y), ∀x, y ∈ R

. The group SP(n, R) can be identiﬁed with the group of

2n ×2n matrices A such that

ωA = ω.

From this equation it follows that det A = ±1, ∀A ∈ SP(n, R). The group

SP(n, R) is called the real symplectic group in 2n dimensions. Let f :

GL(2n, R) → GL(2n, R) be the map deﬁned by

f(A)=A

ωA.

The map f is smooth and hence f

−1

({ω}) is closed. Thus, SP(n, R)=

−1

({ω}) isaclosedsubgroupofGL(2n, R) and thus is a Lie subgroup of

GL(2n, R). Proceeding as in the previous example, we ﬁnd that the Lie alge-

bra of SP(n, R), denoted by sp(n, R), is given by

sp(n, R)={B ∈ M (2n, R) | B

ω = −ωB}.

This also implies that the dimension of SP(n, R) is n(2n +1). The group

SP(n, R) is non-compact. Analogously, one has the complex symplectic

group SP(n, C), which is a Lie group of dimension 2n(2n +1).

As a preparation for the next example we start by brieﬂy recalling the

notations and elementary properties of quaternions. The noncommutative

ring of quaternions is denoted by H in honor of its discoverer, Hamilton (see

104 3 Manifolds

Appendix B for some historical remarks). It is obtained by adjoining to the

ﬁeld R the elements i, j, k satisfying the relations

= j

= k

= −1,

ij = −ji = k, jk = −kj = i, ki = −ik = j.

Thus, H is a 4-dimensional real vector space with basis 1,i,j,k and a quater-

nion x ∈ H has the following expression

x = x

+ ix

+ jx

+ kx

∈ R, 0 ≤ i ≤ 3.

The operation of conjugation on H is deﬁned by

x = x

+ ix

+ jx

+ kx

→ ¯x := x

− ix

− jx

− kx

Then

(xy)=¯y¯x and each nonzero x ∈ H has a unique inverse x

−1

given by

−1

=¯x/(x¯x).

We denote by H

themoduleoverH of n-tuples of quaternions with the

action given by right multiplication. It is customary to refer to a morphism

of this module as an H-linear, or simply a linear map.

Example 3.15 We denote by GL(n, H) the group of linear bijections of H

H can be identiﬁed with C

by the map φ : H → C

deﬁned by

φ(x

+ ix

+ jx

+ kx

)=(x

+ ix

Then GL(n, H) can be identiﬁed with the subgroup of GL(2n, C) of the ma-

trices A of the form

A =



B −



, (3.20)

where B, C are complex n ×n matrices. The group GL(n, H) is a Lie group.

Let us denote by Q the quadratic form on H

deﬁned by

Q(x

,...,x

)=x

+ x

+ ···+ x

, ∀x

,...,x

∈ H.

We deﬁne the Lie subgroup Sp(n) of GL(n, H) by

Sp(n):={A ∈ GL(n, H) | Q(Ax)=Q(x), ∀x ∈ H

The group Sp(n) is called the quaternionic symplectic group in n di-

mensions. This group can be identiﬁed with the group of the 2n ×2n complex

matrices A of the form given in equation (3.20) that satisfy the conditions

†

A =1,A

ωA = ω,

3.6 Lie Groups 105

where ω is the matrix of the canonical symplectic form of the previous exam-

ple. Thus, Sp(n)

∼

U(2n)∩SP(n, C) and is therefore also called the unitary

symplectic group. In particular we have that Sp(1)

∼

SU(2).

The classic reference for the theory of Lie groups is Chevalley [77]; a mod-

ern reference with geometric analysis is Helgason [188].

Chapter 4

Bundles and Connections

4.1 Introduction

In 1931 Hopf studied the set [S

] of homotopy classes of maps of spheres

in his computation of π

). He showed that π

) is generated by the class

of a certain map that is now well known as the Hopf ﬁbration (see Example

4.5). This ﬁbration decomposes S

into subspaces homeomorphic to S

and

the space of these subsets is precisely the sphere S

. In 1933 Seifert introduced

the term ﬁber space to describe this general situation. The product of two

topological spaces is trivially a ﬁber space, but the example of the Hopf

ﬁbration shows that a ﬁber space need not be a global topological product.

It continues to be a local product and is now referred to as a ﬁber bundle.

Fiber spaces immediately acquired an important role in mathematics

through the early work of Whitney, Serre, and Ehresmann. In 1935 Hopf

generalized his early work to construct the ﬁbration of S

over S

with ﬁber

and, using octonions, also constructed a ﬁbration of S

over S

with ﬁber

. All of these examples have found remarkable application to fundamental

solutions of gauge ﬁeld equations (see Chapter 8 for details). For the applica-

tions we wish to consider, we mainly need vector bundles and principal and

associated bundles with a ﬁxed structure group. In Section 4.2 we give several

alternative deﬁnitions of principal bundle and discuss the reduction and

extension of the structure group of the bundle. In physical applications it is

the structure group of the bundle, which serves as the local symmetry

group or the gauge group, while the base is usually a space-time mani-

fold. The matter ﬁelds interacting with gauge ﬁelds are sections of vector

bundles associated to the given principal bundle. The associated bundles are

considered in Section 4.3, where the bundles Ad(P )andad(P ), which play

a fundamental role in gauge theory, are deﬁned. In Section 4.4, we give sev-

eral deﬁnitions of connection on a principal bundle and deﬁne the curvature

2-form. The structure equations and Bianchi identities satisﬁed by the cur-

vature are also given there. Subsection 4.4.1 is devoted to a discussion of

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 4, 107

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