Marathe K. Topics in Physical Mathematics

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108 4 Bundles and Connections

universal connections. The covariant derivative is introduced in Section 4.5,

where the notion of parallel displacement is also discussed. The important

class of linear connections, which include the widely used pseudo-Riemannian

connections, is discussed in Section 4.6. The special case of dimension 4 leads

to the deﬁnitions of various special types of manifolds that play a signiﬁcant

role in physical applications. A section on generalized connections concludes

the chapter.

A standard reference for most of the material in this chapter is Kobayashi

and Nomizu [225]. The theory of ﬁber bundles occupies a central place in

modern mathematics. Recently, ﬁber bundles have provided the geometrical

setting for studying gauge ﬁeld theories. We discuss various basic aspects of

ﬁeld theories in Chapters 6 and 7. Standard references for the theory of ﬁber

bundles are Husem¨oller [198] and Steenrod [359].

4.2 Principal Bundles

We begin with the deﬁnition of a diﬀerentiable ﬁber bundle.

Deﬁnition 4.1 A diﬀerentiable ﬁber bundle E over B is a quadruple

ζ =(E, B,π,F),whereE,B,F are diﬀerentiable manifolds and the map

π : E −→ B is an open diﬀerentiable surjection satisfying the following local

triviality (LT) property:

LT property—There exists an open covering {U

}

i∈I

of B and a family

of diﬀeomorphisms ψ

: U

× F −→ π

−1

), ∀i ∈ I, satisfying the

condition (π ◦ψ

)(x, g)=x, ∀(x, g) ∈ U

× F.

The family {(U

,ψ

)}

i∈I

is called a local coordinate representation or

a local trivialization of the bundle ζ. E is called the total space or the

bundle space, B the base space, π the bundle projection of E on B,

and F the standard or typical ﬁber. E

:= π

−1

(x) is called the ﬁber of

E over x ∈ B.

The bundle τ =(B × F, B, π, F ), where π is the projection onto the ﬁrst

factor, is called the trivial bundle over B with ﬁber F . The total space

of the trivial bundle τ is B × F. For this reason a general ﬁber bundle is

sometimes called a twisted product of B (the base) and F (the standard

ﬁber). A ﬁber bundle ζ =(E, B,π,F) is sometimes indicated by a diagram

as follows:

4.2 Principal Bundles 109

One can verify that, ∀x ∈ U

, ∀i ∈ I,themapψ

i,x

: F → E

deﬁned by

g → ψ

(x, g) is a diﬀeomorphism. If U

∩U

= ∅ then we write U

= U

∩U

and deﬁne

: U

→ Diﬀ(F )byψ

(x)=ψ

−1

i,x

◦ ψ

j,x

The functions ψ

are called the transition functions for the local repre-

sentation. They satisfy the condition

(x) ◦ ψ

(x) ◦ψ

(x)=id

, ∀x ∈ U

ijk

, (4.1)

where we have written U

ijk

= U

∩U

. Condition (4.1)isreferredtoasthe

cocycle condition on the transition functions. We note that, given the base

B, the typical ﬁber F and a family of transition functions {ψ

} satisfying the

cocycle condition (4.1), it is possible to construct the ﬁber bundle E. Given

the bundles ζ =(E, B,π,F)andζ



=(E



,π



), a bundle morphism

f from ζ to ζ



is a diﬀerentiable map f : E → E



such that f maps the ﬁbers

of E smoothly to the ﬁbers of E



and therefore induces a smooth map of B

to B



denoted by f

. Thus, we have the following commutative diagram:



If B = B



, f is injective and f

= id

,thenwesaythatζ is a subbundle

of ζ



.Ifζ =(E, B,π,F) is a ﬁber bundle, we frequently denote by E the

ﬁber bundle ζ when B,π,F are understood. If ζ =(E, B,π,F)isaﬁber

bundle and h ∈F(M,B), where M is a manifold, then we can deﬁne a

bundle h

∗

ζ =(h

∗

E,M, h

∗

π, F) called the pull-back of the bundle E to M

as follows. h

∗

E is the subset of M ×E consisting of the pairs (p, a) ∈ M ×E

such that h(p)=π(a), and one can show that it is a closed submanifold

of M × E.Thus,h

∗

E is obtained by attaching to each point p ∈ M the

ﬁber π

−1

(h(p)) over h(p). The map h

∗

π is the restriction to h

∗

E of the

natural projection of M × E onto M.Themaph lifts to a unique bundle

map

h : h

∗

E → E such that the following diagram commutes:

∗

Let ζ =(E, B,π,F) be a ﬁber bundle. A smooth map s : B → E with

π ◦ s = id

is called a (smooth) section of the ﬁber bundle E over B.We

110 4 Bundles and Connections

denote by Γ (B, E), or simply Γ (E), the space of sections of E over B.If

U ⊂ B is open then we denote by Γ (E

) the set of sections of the bundle

E restricted to U.Ifp ∈ U and s ∈ Γ (E

)thenwesaythats is a local

section of E at p.

Choosing local coordinates {x

| 1 ≤ i ≤ m} in a neighborhood of p ∈ B

and {(x

)| 1 ≤ i ≤ m, 1 ≤ j ≤ n} in a neighborhood of s(p) ∈ E,wemay

think of s as a function from an open subset of R

to R

m+n

such that

) → (x

,...,x

)).

The Taylor expansion of s at p clearly depends on the local coordinates

chosen. However, if two local sections s and t have the same kth order Taylor

expansion at p in one coordinate system, then they have the same kth order

expansion in any other coordinate system. This observation can be used to

deﬁne an equivalence relation on local sections at p. An equivalence class

determined at p by the section s is called the k-jet of s at p and is denoted

by j

(s)

. We deﬁne the k-jet of sections of E over B, J

(E/B)(also

denoted by J

(E)) by

(E/B):={j

(s)

| p ∈ B, s is a local section of E at p}.

(E/B) is a ﬁber bundle over B with projection π

: J

(E) → B deﬁned by

(s)

)=p, and a ﬁber bundle over J

(E/B), 0 ≤ l ≤ k, with projection

: J

(E) → J

(E) deﬁned by π

(s)

)=j

(s)

.Inparticular,J

(E)

is a ﬁber bundle over J

(E)=E. A section s ∈ Γ (E) induces a section

(s) ∈ Γ (J

(E)) deﬁned by j

(s)(p)=j

(s)

.Wecallj

(s)thek-jet

extension of s.Themapj

: Γ (E) → Γ (J

(E)), deﬁned by

s → j

(s),

is called the k-jet extension map.IfM, N are manifolds then we deﬁne the

space J

(M,N)ofk-jets of maps of M to N by

(M,N)=J

((M × N)/M ),

where M × N is regarded as a trivial ﬁber bundle over M.Itispossibleto

deﬁne the bundle J

(M,N) directly by considering the Taylor expansion of

local maps up to order k. Jet bundles play a fundamental role in the geo-

metrical formulation of variational problems and in particular of Lagrangian

theories. We shall not make use of these jet bundle techniques in this book.

Let ζ =(E,B,π,F) be a ﬁber bundle. If a Lie group G is a subgroup of

Diﬀ(F ) such that for each transition function ψ

of ζ we have ψ

(x) ∈ G

∀x ∈ U

,andψ

is a smooth map of U

into G,thenwesaythatζ is a ﬁber

bundle with structure group G.Wenowgivethetwomostimportant

cases of ﬁber bundles with structure group.

4.2 Principal Bundles 111

Deﬁnition 4.2 Aﬁberbundleζ =(E,B,π,F) with structure group G is

called a vector bundle with ﬁber type F if F is a Banach space and G is

the Lie group of the linear automorphisms (linear continuous bijections) of

F .Inparticular,ifF is a real (resp., complex) vector space of dimension

n and G = GL(n, R) (resp., G = GL(n, C)), then we call ζ a real (resp.,

complex) vector bundle of rank n.

We note that in the case of a vector bundle of rank n the transition func-

tions turn out to be automatically smooth. Let E, H be two vector bundles

over B. The algebraic operations on vector spaces can be extended to deﬁne

vector bundles such as E⊕H, E⊗H,andHom(E,H) by using pointwise oper-

ations on the ﬁbers over B. In particular, we can form the bundle (A

B)⊗E.

The sections of this bundle are called k-forms on B with values in the vec-

tor bundle E,orsimply,vector bundle-valued (E-valued) k-forms . We

write Λ

(B,E) for the space of sections Γ ((A

B) ⊗E). Thus, α ∈ Λ

(B,E)

can be regarded as deﬁning for each x ∈ B a k-linear, anti-symmetric map

of (T

into E

.Inparticular,Λ

(B,E)=Γ (E). If E is a trivial vector

bundle with ﬁber V ,thenwecallΛ

(B,E)thespaceofk-forms with values

in the vector space V , or vector valued (V -valued) k-forms, and denote it by

(B,V ). We observe that a Riemannian metric on M is a smooth section

of the vector bundle S

(TM)onM, i.e., of the bundle whose ﬁber on x ∈ M

is the vector space S

M) of symmetric bilinear maps of T

M ×T

M into

R. More generally, given the real (resp., complex ) vector bundle E on M,a

Riemannian (resp., Hermitian) metric on E is a smooth section s of the

vector bundle S

(E)onM such that s(x) is a bilinear (resp., sesquilinear),

symmetric (resp., Hermitian) and positive deﬁnite map of E

× E

into R

(resp., C), ∀x ∈ M. By deﬁnition a Riemannian metric on M is a Riemannian

metric on the vector bundle TM on M .ARiemannian (resp., Hermitian)

vector bundle is a couple (E,s)whereE is a real (resp., complex) vector

bundle and s is a Riemannian (resp., Hermitian) metric on E.Wenotethat

if M is paracompact, then every real (resp., complex) vector bundle on M

admits a Riemannian (resp., Hermitian) metric.

Let V

be vector spaces and h : V

× V

→ V

be a bilinear form.

Let α ∈ Λ

(B,V

),β∈ Λ

(B,V

); then we deﬁne α ∧

β ∈ Λ

p+q

(B,V

)as

follows. Let {u

} be a basis of V

and {v

} abasisforV

.Thenwecanwrite

α = α

,β= β

where α

∈ Λ

(B)andβ

∈ Λ

(B) and we deﬁne

α ∧

β := α

∧ β

h(u

There are several important special cases of this operation. For example, if

V = V

= V

and if h is an inner product on the vector space V ,then

α ∧

β ∈ Λ

p+q

(B). If V is a Lie algebra and h is the Lie bracket, then it is

customary to denote α ∧

β by [α, β]. Thus, we have

112 4 Bundles and Connections

[α, β]:=α

∧ β

]=c

∧ β

where c

are the structure constants of the Lie algebra V .IfB is a pseudo-

Riemannian manifold with metric g and h is an inner product on V ,thenfor

each x ∈ B,thespaceΛ

(B,V ) becomes a pseudo-inner product space, with

pseudo-inner product denoted by  , 

(g,h)

, deﬁned by

α, β

(g,h)

:= α

,β



h(u

),α,β∈ Λ

(B,V ),

where <, >

is the pseudo-inner product on Λ

(B) induced by g.IfB is

compact with volume form v

, then we can integrate these local products

over B to obtain a pseudo-inner product on Λ

(B,V ) deﬁned by

α, β

(g,h)



α

,β



h(u

)dv

,α,β∈ Λ

(B,V ).

Recall that a natural symmetric bilinear form on a Lie algebra V is given

by the Killing form

K(X, Y ):=Tr(adX ◦ad Y ), ∀X, Y ∈ V,

where Tr(L) denotes the trace of the linear transformation L of V .Wenote

that, if c

are the structure constants of V with respect to a basis E

,then

K(X, Y )=c

Furthermore, K is non-degenerate if and only if the determinant of the matrix

} is nonzero. By Cartan’s criterion K is non-degenerate if and only

if V is semisimple, i.e., if it has no nonzero Abelian ideals. If V = g is the Lie

algebra of a connected Lie group G and g is semisimple, then, by a theorem

of Weyl, K is negative deﬁnite if and only if G is compact. Thus, in this case

h := −K is an inner product on g which with h the Killing form and g acan

be used to deﬁne the norm or energy of α ∈ Λ

(B,V )by

α :=



|α, α

(g,h)

The constructions discussed above for vector-valued forms can be extended

to apply to vector bundle-valued forms by using their pointwise vector space

structures. We shall use them to deﬁne the Yang–Mills and other action

functionals in gauge theories.

Example 4.1 Let TM be the tangent space of the m-dimensional manifold

M and π : TM → M the canonical projection. Then (TM,M,π,R

) is a

vector bundle of rank m, called the tangent bundle of M.Ak-dimensional

distribution on M is a vector bundle of rank k over M , which is a subbundle

of the tangent bundle of M .

4.2 Principal Bundles 113

Deﬁnition 4.3 Aﬁberbundleζ =(P, M, π, F ) with structure group G is

called a principal ﬁber bundle,orsimplyaprincipal bundle, over M

with structure group G if F is a Lie group and G is the Lie group of the

diﬀeomorphisms h of F such that

h(g

)=h(g

The map h → h(e), where e is the identity of F , is an isomorphism of G with

F . A principal bundle over M with structure group G is denoted by P (M, G).

Equivalently a principal bundle P (M,G) with structure group G over M

may be deﬁned as follows.

Deﬁnition 4.4 A principal bundle P (M,G) with structure group G over M

is a ﬁber bundle (P, M, π, G) with a free right action ρ of G on P , such that

1. the orbits of ρ are the ﬁbers of π : P → M, i.e., π may be identiﬁed with

the canonical projection P → P/G;

2. ∀ψ : U × G → π

−1

(U), such that ψ is a local trivialization of P ,writing

g in the place of ρ(u

,g), one has

−1

g)=ψ

−1

)g, ∀u

∈ P

,g∈ G.

We observe that, given the ﬁrst deﬁnition one has the natural right action

ρ of G on P deﬁned by

ρ(u

,g)=ψ

i,x

(ψ

−1

i,x

)g),

where ψ

is a local trivialization of P at x ∈ M.Thisactionρ is a free right

action and satisﬁes both conditions of Deﬁnition 4.4. Conversely, given the

second deﬁnition, the ψ

(x) are diﬀeomorphisms of G satisfying the deﬁning

condition

(x)(g

)=ψ

(x)(g

of Deﬁnition 4.3. We give below an example of a principal bundle that is

naturally associated with every manifold.

Example 4.2 (Bundle of frames) Let M be an m-dimensional manifold. A

frame u =(u

,...,u

) at a point x ∈ M is an ordered basis of the tangent

space T

M.Let

(M)={u | u is a frame at x ∈ M},

L(M)=



x∈M

(M).

Deﬁne the projection

π : L(M) → Mbyu→ x, where u ∈ L

(M) ⊂ L(M ).

114 4 Bundles and Connections

The general linear group GL(m, R) acts freely on L(M ) on the right by

(u, g) → ug =(u

,...,u

),g=(g

) ∈ GL(m, R).

It can be shown that L(M) can be given the structure of a manifold, such that

this GL(m, R)-action is smooth. Then L(M)(M,GL(m, R)) is a principal

bundle over M with structure group GL(m, R). This principal bundle L(M)

is called the bundle of frames or simply the frame bundle of M .

Let P (M,G)andQ(N,H) be two principal ﬁber bundles. A principal

bundle homomorphism of Q(N, H)intoP (M, G) is a bundle homomor-

phism f : Q → P together with a Lie group homomorphism γ : H → G,such

that

f(uh)=f(u)γ(h), ∀u ∈ Q, h ∈ H.

Whe

n f s

atisﬁes the above condition we say that it is equivariant with

respect to γ. A bundle isomorphism f : P → P m which is equivariant with

respect to the identity is called a principal bundle automorphism.The

set of all principal bundle automorphisms of P is a subgroup of Diﬀ(P ),

called the automorphism group of P . It is denoted by Aut(P ). Returning

to the general case, if f : Q → P is an embedding and γ is injective, then we

say that Q is embedded in P . Note that in this case the induced morphism

: N → M is also an embedding. If M = N and f

= id

, then Q

is called a reduced subbundle of P or a reduction of the structure

group G to H where H is regarded as a subgroup of G. The structure group

G of the principal bundle P (M,G)issaidtobereducible to a subgroup

H ⊂ G if P (M, G) admits a reduction of G to H.IfH is a maximal compact

subgroup of G, then it can be shown that the bundle P (M,G) can be reduced

to a bundle Q(M, H). An application of this result to the bundle of frames

L(M)(Example(4.2)) shows that L(M)(M,GL(m, R)) can be reduced to a

subbundle O(M ) with structure group O(m, R), the orthogonal group. The

bundle O(M)(M, O(m, R)) is called the bundle of orthonormal frames on

M. Furthermore, this reduction is equivalent to the existence of a Riemannian

structure on M. The structure group O(m, R)ofO(M)

can be reduced to

he special orthogonal group SO(m, R) if and only if the manifold M is

orientable. The reduced subbundle of special orthonormal frames is denoted

by SO(M)(M, SO(m, R)).

On the other hand, in some situations one is interested in extending,or

lifting, the structure group of a bundle to obtain a new principal bundle in

the following sense. Let P (M,G) be a principal bundle with structure group

G. Recall that the center of a group H is the normal subgroup Z(H)ofH

given by

Z(H):={x ∈ H | xh = hx, ∀h ∈ H}.

Let H be a Lie group and let f : H → G be a surjective, covering homomor-

phism such that K =Kerf ⊂ Z(H), the center of H.WesaythatP (M, G)

has a lift to a principal bundle Q(M, H) with structure group H if there

4.2 Principal Bundles 115

exists a bundle map

f : Q → P such that

f(uh)=

f(u)f (h), ∀u ∈ Q, h ∈ H.

Using

Cech cohomology, Greub and Petry [165] have shown that there exists

a topological obstruction η(P ) ∈ H

(M,K) to the lifting of P (M,G)to

Q(M,H) with the property that

h ∈F(N,M) implies that η(h

∗

(P )) = h

∗

(η(P )).

An important special case that appears in many applications is when f

is the universal covering map. In this case K is isomorphic to π

(G), the

fundamental group of G, and hence η(P ) ∈ H

(M,π

(G)). When f is the

universal covering map we say that Q(M,H)isauniversal lift,orauni-

versal extension,ofP (M, G). The following theorem gives the obstruction

η(P ) in terms of well known characteristic classes (see Chapter 5) for three

frequently used groups.

Theorem 4.1 In the following, H denotes the universal covering group of G.

1. Let G = SO(n),thenH = Spin(n). In this case the obstruction η(P )=

(P ) ∈ H

(M,Z

),wherew

(P ) is the second Stiefel–Whitney class of

P .

2. Let G = SO(3, 1)

, the connected component of the identity of the proper

Lorentz group; then H = Spin(3, 1) = SL(2, C). In this case η(P )=

(P ) ∈ H

(M,Z

),wherew

(P ) is the second Stiefel–Whitney class of

P .

3. Let G = U (n),thenH = R × SU(n). In this case η(P )=c

(P ) ∈

(M,Z),wherec

(P ) is the ﬁrst Chern class of P .

The following example is a typical application of the above theorem.

Example 4.3 Let M be an oriented Riemannian m-manifold. Then its bun-

dle of special orthonormal frames SO(M) is a principal SO(m, R) bundle.

We say that M admits a spin structure if the bundle SO(M) can be ex-

tended to the group Spin(m, R). By Theorem 4.1 it follows that M admits a

spin structure if and only if the second Stiefel–Whitney class of M is zero.

A spin manifold is a manifold M together with a ﬁxed spin structure. The

principal bundle obtained by this extension is called the spin frame bundle

and is denoted by Spin(M)(M,Spin(m, R)),orsimplybySpin(M).

The deﬁnition of spin manifold can be extended to an oriented pseudo-

Riemannian manifold. In the physical literature one usually deals with the

case of a 4-dimensional Lorentz manifold. Item 2 of Theorem 4.1 refers to

this case.

Let P (M,G) be a principal bundle. The action ρ of G on P deﬁnes for

each u ∈ P the map ρ

: G → P by ρ

(g)=ρ(u, g). This induces an injective

homomorphism of the Lie algebra g of G into X(P ) (the Lie algebra of the

116 4 Bundles and Connections

vector ﬁelds on P ) as follows. Let A ∈ g and let a

= exp(tA)betheone-

parameter subgroup of G generated by A. Restricting the action ρ of G to

exp(tA)wegetasmoothcurve

u ·exp(tA):=ρ

(exp(tA)) = ρ(u, exp(tA))

through u ∈ P . The tangent vector to this curve at the point u ∈ P is denoted

.Thefundamental vector ﬁeld

A ∈X(P )ofA ∈ g is deﬁned by

the map

u →

:= Tρ

(A).

One can verify that the map ˜ρ from g to X(P ) deﬁned by

A →

is an injective Lie algebra homomorphism. We note that, since G acts freely

on P , A = 0 implies

=0, ∀u ∈ P .Moreover,

A has another important

property, namely, that it is a vertical vector ﬁeld in the sense of the following

deﬁnition. A vector ﬁeld Y ∈X(P )issaidtobeavertical vector ﬁeld if

is tangent to the ﬁber of P through u, ∀u ∈ P .WedenotebyV(P )the

set of vertical vector ﬁelds on P .ThesetV(P ) is called the vertical bundle

of P . It is a Lie subalgebra of X(P ). Since G preserves the ﬁbers (i.e., acts

vertically on P ),

A is a vertical vector ﬁeld and the map

˜ρ : g →V(P ) deﬁned by A →

is a Lie algebra isomorphism into V(P ) and induces a trivialization of the

vertical bundle V(P )

∼

P ×g. The inverse of this isomorphism associates to

a vertical vector ﬁeld an element of the Lie algebra g. This result plays an

important role in a deﬁnition of connection given later in this chapter.

4.3 Associated Bundles

Let P (M,G) be a principal ﬁber bundle and F be a left G-manifold, i.e., G

acts by diﬀeomorphisms from the left on F .Wedenotebyr this action of

G on F and by gf or r(g)f the element r(g, f). Then we have the following

right action R of G on the product manifold P × F

(u, f)g =(ug, r(g

−1

)f), ∀g ∈ G, (u, f) ∈ P × F :

The action R is free and its orbit space (P × F )/R is denoted by P ×

or by E(M,F,r,P)(orsimplyE). We denote by

O : P × F → E

4.3 Associated Bundles 117

the quotient map, called the projection onto the orbits of R,andby[u, f ]

the orbit O(u, f). We have the following commutative diagram:

P × FP

where π

: E → M is deﬁned by [u, f] → π(u)andp

is the projection onto

the ﬁrst factor. (E,M, π

,F) is a ﬁber bundle over M with ﬁber type F.We

call P ×

F or E(M, F,r,P)theﬁber bundle associated to P with ﬁber

type F .IfF is a vector space then E is called the vector bundle with

ﬁber type F associated to P .

Example 4.4 Let Ad denote the adjoint action of G on itself. Then P ×

is a bundle of Lie groups associated to P , denoted by Ad(P ). We note that

Ad(P ) is not a principal bundle, in general. Let ad denote the adjoint action

of G on its Lie algebra g.ThenP ×

g is a bundle of Lie algebras associated

to P , denoted by ad(P ).ThebundlesAd(P ) and ad(P) play a fundamental

role in applications to gauge theory.

Each u ∈ P induces an isomorphism

˜u : F → E

π(u)

, deﬁned by ˜u(f)=O(u, f ),

where O is the orbit projection. We note that the map ˜u satisﬁes the relation



(ug)(f)=˜u(gf), ∀g ∈ G.

We denote by F

(P, F )thespaceofG-equivariant maps of P to F , i.e.,

(P, F ):={f : P → F | f(ug)=g

−1

(f(u)), ∀g ∈ G}.

Recall that Γ (E)=Γ

(E(M, F,r,P)) is the space of smooth sections of

E over M. There exists a one-to-one correspondence between F

(P, F )and

Γ (E) which is deﬁned as follows: For f ∈F

(P, F ) we deﬁne s

∈ Γ (E)by

(x)=[u, f(u)], where u ∈ π

−1

(x).

It is easy to verify that s

is well deﬁned and that f → s

is a one-to-one

correspondence from F

(P, F )toΓ (E) with the inverse deﬁned as follows:

If s ∈ Γ (E), then f

∈F

(P, F ) is deﬁned by

(u)=˜u

−1

(s(π(u))).

Example 4.5 Let M be an m-dimensional manifold. The tangent bundle

TM of M is an associated bundle of L(M ) with ﬁber type R

and left action