Marathe K. Topics in Physical Mathematics

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118 4 Bundles and Connections

r given by the deﬁning representation of GL(m, R) on R

, i.e.,

TM = E(M,R

,r,L(M)).

For u ∈ L

(M) we have the map

˜u : R

→ T

(M),

which is a linear isomorphism. Thus, we could deﬁne a frame by the map ˜u

using the vector bundle structure of TM with the action of GL(m, R) given

by the composition

˜u · g =˜u ◦g, ∀g ∈ GL(m, R),

where g is regarded as a map from R

to R

. Similarly it can be shown that

the various tensor bundles T

(M) and form bundles Λ

(M) are associated

bundles of L(M ).

Example 4.6 Let M be a spin manifold with spin bundle Spin(M).Let

Spin(M ) ×

V be the bundle associated to Spin(M) by the representation ρ

of Spin(m, R) on the complex vector space V .ThenS

= Γ (Spin(M ) ×

V )

is called the space of spinors of type ρ. By the ﬁrst part of the Theo-

rem 4.1 we can conclude that M admits a spin structure if and only if

(M):=w

(SO(M )), the second Stiefel–Whitney class of M,iszero.Topo-

logical classiﬁcation of spin structures is attributed to Milnor. A detailed ac-

count of spin geometry may be found in Lawson and Michelsohn [248]. The

mathematical foundations of the theory of spinors over Lorentz manifolds

were laid by Cartan in [71], where the Dirac operator on spinors was intro-

duced to study Dirac’s equation for the electron. This operator and its various

extensions play a fundamental role in the study of the topology and the ge-

ometry of manifolds arising in gauge theory.

Associated bundles can be used to give the following formulation of the

concept of reduction of the structure group of a principal bundle.

Theorem 4.2 If H isaclosedsubgroupofG, then the structure group

G of P (M, G) is reducible to H if and only if the associated bundle

E(M,G/H,q,P(M,G)) admits a section (where q is the canonical action

of G on the quotient G/H).

This theorem is useful in studying the existence of pseudo-Riemannian

structures on a manifold and their topological implications.

4.4 Connections and Curvature 119

4.4 Connections and Curvature

The idea of Riemannian manifold was introduced in Riemann’s famous lec-

ture “

Uber die Hypothesen, welche der Geometrie zugrunde liegen” (On the

hypotheses that lie at the foundations of geometry). Riemann’s ideas were

extended by Ricci and Levi-Civita, who gave a systematic account of Rie-

mannian geometry and also introduced the notion of covariant diﬀerentiation.

Their work inﬂuenced Einstein (via Grossmann), who used it to formulate his

general theory of relativity. Weyl and Cartan introduced the idea of aﬃne,

projective, and conformal connections and found interesting applications of

these to physical theories. The notion of connection in a ﬁber bundle was

introduced by Ehresmann and this inﬂuenced all subsequent developments

of the theory of connections. In particular his notion of a Cartan connection

includes aﬃne, projective, and conformal connections as special cases.

In this section we give several deﬁnitions of connection on a principal

bundle and deﬁne the curvature 2-form. The structure equations and Bianchi

identities satisﬁed by the curvature are also given there. A subsection is

devoted to a detailed discussion of universal connections.

Let P (M,G) be a principal bundle with structure group G and canonical

projection π over a manifold M of dimension m. Recall that a k-dimensional

distribution on P is a smooth map L : P → TP such that L(u)isak-

dimensional subspace of T

P , for all u ∈ P .

Deﬁnition 4.5 A connection Γ in P (M,G) is an m-dimensional distribu-

tion H : u → H

P ,onP such that the following conditions are satisﬁed for

all u ∈ P :

1. T

P = V

P ⊕ H

P ,whereV

P =Ker(π

∗u

) is the vertical subspace of

the tangent space T

P ;

2. H

(u)

P =(ρ

)

∗

P, ∀a ∈ G, where ρ

is the right action of G on P

determined by a.

The ﬁrst condition in the above deﬁnition can be taken as a deﬁnition of

a horizontal distribution H. The second condition may be rephrased as

follows:

2’. The distribution H is invariant under the action of G on P .

We call H

P (also denoted simply by H

)theΓ -horizontal subspace,

or simply the horizontal subspace of T

P . The union HP of the horizontal

subspaces is a manifold, called the horizontal bundle of P. The union VP

of the vertical subspaces is a manifold called the vertical bundle of P .We

will also write simply V

instead of V

P . A vector ﬁeld X ∈X(P ) is called

vertical if X(u) ∈ V

P, ∀u ∈ P . We note that while the deﬁnition of HP

depends on the connection on P , the deﬁnition of the vertical bundle VP is

independent of the connection on P . Condition (1) allows us to decompose

each X ∈ T

P into its vertical part v(X) ∈ V

and the horizontal part

h(X) ∈ H

.IfY ∈X(P ) is a vector ﬁeld on P then

120 4 Bundles and Connections

v(Y ):P → TP deﬁned by u → v(Y

)

and

h(Y ):P → TP deﬁned by u → h(Y

)

are also in X(P ). We observe that

∗u

: H

→ T

π(u)

is an isomorphism. The vector ﬁeld Y ∈X(P )issaidtobeΓ -horizontal

(or simply horizontal) if Y

∈ H

, ∀u ∈ P . The set of horizontal vector ﬁelds

is a vector subspace but not a Lie subalgebra of X(P ). For X ∈X(M)the

Γ -horizontal lift (or simply the lift)ofX to P is the unique horizontal

ﬁeld X

∈X(P ) such that π

∗

= X.NotethatX

∈X(P ) is invariant

under the action of G on P and every horizontal vector ﬁeld Y ∈X(P )

invariant under the action of G is the lift of some X ∈X(M), i.e., Y = X

for some X ∈X(M). In the following proposition we collect some important

properties of the horizontal lift of vector ﬁelds.

Proposition 4.3 Let P (M,G) be a principal bundle with connection Γ .Let

X, Y ∈X(M) and f ∈F(M); then we have the following properties:

1. X

+ Y

=(X + Y )

2. (π

∗

f)X

=(fX)

3. h([X

]) = [X,Y]

Asmoothcurvec in P (i.e. c is a smooth function from some open interval

I ⊂ R into P ) is called a horizontal curve if ˙c(t) ∈ H

c(t)

, ∀t ∈ I. A section

s ∈ Γ (P ) is called parallel if

∗

M) ⊂ H

s(x)

, ∀x ∈ M ;

i.e., if s ◦ c is a horizontal curve for all curves c in M.Givenacurve

x :[0, 1] → M and w

∈ P such that π(w

)=x(0), there is a unique

horizontal curve w :[0, 1] → P such that w(0) = w

and

π(w(t)) = x(t), ∀t ∈ [0, 1].

The curve w in P is called the horizontal lift of the curve x starting from

∈ P. If P

(resp., P

) denotes the ﬁber of P over x(0) (resp., x(1)) then

the horizontal lift of x to P induces a diﬀeomorphism of P

with P

called the

parallel displacement along x. In particular, given a loop (closed curve)

x at p ∈ M, i.e., p = x

= x

, the horizontal lift of x to P induces an

automorphism of the ﬁber π

−1

(p). The set of all such automorphisms forms

a group called the holonomy group of the connection Γ at p ∈ M .Itis

denoted by Φ(p). The subset of Φ(p) corresponding to parallel displacement

along loops homotopic to the constant loop at p turns out to be a subgroup

of Φ(p). It is called the restricted holonomy group of the connection Γ

4.4 Connections and Curvature 121

at p ∈ M and is denoted by Φ

(p). Given a point u ∈ π

−1

(p), each α ∈ Φ(p)

(resp., Φ

(p)) determines a unique g ∈ G such that α(u)=ug.Themapα →

g of Φ(p)(resp.,Φ

(p)) to G is an isomorphism onto a subgroup Φ

(resp.,

) called the holonomy group (resp., restricted holonomy group)of

the connection Γ at u ∈ P .ItcanbeshownthatifM is paracompact, then

the holonomy group Φ

is a Lie subgroup of the structure group G, Φ

is a

normal subgroup of H

and Φ

/Φ

is countable. Now we let P (u)denotethe

set of points of P that can be joined to u by a horizontal curve. Then we

have the following theorem.

Theorem 4.4 Let M be a connected, paracompact manifold and Γ acon-

nectionintheprincipalbundleP (M,G).Then

1. P (u),u∈ P , is a reduced subbundle of P with structure group Φ

.Itis

called the holonomy bundle (of Γ ) through u.

2. The holonomy bundles P (u),u∈ P , are isomorphic with one another and

partition P into a disjoint union of reduced subbundles.

3. The connection Γ is reducible to a connection in P (u).

We now give two important characterizations of a connection that are often

used as deﬁnitions of a connection. Recall ﬁrst that a k-form α ∈ Λ

(P, V )

(the space of k-forms on P with values in the vector space V )canbeidentiﬁed

with the map

u → α

,u∈ P,

where

: T

P ×···×T



 

k times

→ V

is a multilinear anti-symmetric map. Given a basis {v

}

1≤i≤n

in V ,wecan

express α as the formal sum

α =



i=1

where α

∈ Λ

(P ), ∀i. We deﬁne a 1-form ω ∈ Λ

(P, g)onP with values in

the Lie algebra g by using the connection Γ as follows:

)=˜ρ

−1

(v(X

)), ∀u ∈ P, ∀X

∈ T

P, (4.2)

where ˜ρ

: g →V(P )

is the isomorphism induced by the action of G on P .

It is customary to write (4.2)intheform

ω(X)=˜ρ

−1

(v(X)). (4.3)

The 1-form ω is called the connection 1-form of the connection Γ.Itcan

be shown that the connection 1-form ω satisﬁes the following conditions:

ω(

A)=A, ∀A ∈ g, (4.4)

122 4 Bundles and Connections

(ρ

)

∗

ω =ad(a

−1

)ω, ∀a ∈ G. (4.5)

Condition (4.5) means that ω is G-equivariant, i.e., the following diagram

commutes:

ad(a

−1

)

Tρ

Condition (4.5) can also be expressed as

ω(ua)(Tρ

(X(u))) = ad(a

−1

)(ω(u)X(u)), ∀X ∈X(P ),

wherewehavewrittenX(u)forX

etc. Conditions (4.4) and (4.5) char-

acterize the connection 1-form ω on P . In fact, one may give the following

alternative deﬁnition of a connection.

Deﬁnition 4.6 A connection in P (M, G) is a 1-form ω on P with values

in the Lie algebra g) (i.e., ω ∈ Λ

(P, g)), which satisﬁes conditions (4.4) and

(4.5) given above.

Note that given a 1-form ω satisfying conditions (4.4) and (4.5), we can

deﬁne the distribution H : u → H

on P by

:= {Y ∈ T

P | ω

(Y )=0}. (4.6)

One can then verify that the distribution H is m-dimensional and deﬁnes

a connection Γ according to the Deﬁnition 4.5 and that the connection 1-

form associated to Γ is ω.Leth be a Lie subalgebra of g. We say that the

connection ω is reducible if ω(X) ∈ h, ∀X ∈X(P ). A connection ω that is

not reducible is called irreducible. These concepts play an important role

in studying the space of all connections on P. These spaces are fundamental

in the construction of various gauge ﬁelds that we study in later chapters.

We now motivate another characterization of a connection in terms of a lo-

cal representation of the bundle P (M, G). Let ω be a connection on P (M,G).

Let {(U

,ψ

)}

i∈I

be a local representation of P (M,G) with transition func-

tions

: U

→ G.

Let e ∈ G be the identity element of G and let s

: U

→ P be a local section

deﬁned by

(x)=ψ

(x, e).

Deﬁne the family {ω

}

i∈I

of 1-forms

∈ Λ

, g)byω

= s

∗

(ω),

4.4 Connections and Curvature 123

where ω is a connection on P .LetΘ ∈ Λ

(G, g) be the canonical left invariant

1-form on G deﬁned by Θ(A)=A, ∀A ∈ g. Alternatively, Θ can be deﬁned

Θ(g):=TL

−1

: T

G → T

G ≡ g, ∀g ∈ G.

Then writing Θ

= ψ

∗

Θ, we obtain the following relations

(x)=ad(ψ

(x)

−1

)ω

(x)+Θ

(x), ∀x ∈ U

and ∀i, j ∈ I. (4.7)

Thus, given a connection ω on P ,wehaveafamily{(U

,ψ

,ω

)}

i∈I

where

{(U

,ψ

)}

i∈I

is a local representation of P and {ω

}

i∈I

is a family of 1-forms

satisfying conditions (4.7). Conversely, given such a family {(U

,ψ

,ω

)}

i∈I

satisfying conditions (4.7), a connection is determined in the following way.

Let ψ

−1

: π

−1

) → U

× G and let p

be the canonical projections of

× G on the ﬁrst and second factors, respectively. Let

=(p

◦ ψ

−1

)

∗

+(p

◦ ψ

−1

)

∗

Θ.

Then α

∈ Λ

(π

−1

), g). Deﬁne ω ∈ Λ

(P, g)byω = α

on π

−1

). This

is well deﬁned because on the intersection π

−1

) the conditions guarantee

that α

= α

. Then one can show that ω is a connection according to Def-

inition 4.6. Thus we may give the following third deﬁnition of a connection

equivalent to the two deﬁnitions given above.

Deﬁnition 4.7 A connection in the bundle P (M,G) is a family of triples

{(U

,ψ

,ω

)}

i∈I

,where{(U

,ψ

)}

i∈I

is a local representation of P and {ω

∈

, g)}

i∈I

is a family of 1-forms satisfying the relations (4.7).

Frequently, it is this local deﬁnition that is used to construct a connection

via a suitable local representation of P .

Example 4.7 Let M be a paracompact manifold and let P = L(M) be the

bundle of frames on M. Recall that M admits a Riemannian metric so that

thebundleofframesL(M) can be reduced to a bundle of orthonormal frames.

Using the local isomorphism with R

we can pull back the ﬂat connection on

the frame bundle of R

to L(M), locally. These local connections can be

pieced together by a partition of unity argument to obtain the Riemannian

connection in L(M).

Let φ ∈ Λ

(P, V )beak-form in P with values in a ﬁnite-dimensional

vector space V .Letr : G → GL(V ) be a representation of G on V .Wesay

that φ is pseudo-tensorial of type (r, V )if

∗

φ = r(a

−1

) ·φ, ∀a ∈ G.

A connection 1-form ω on P is pseudo-tensorial of type (ad, g). The form

φ ∈ Λ

(P, V ) is called horizontal if

φ(X

,...,X

)=0

124 4 Bundles and Connections

whenever some X

,1≤ i ≤ k, is vertical. The form φ ∈ Λ

(P, V ) is called

tensorial of type (r, V ) if it is horizontal and pseudo-tensorial of type (r, V ).

If the k-form φ ∈ Λ

(P, V ) is tensorial of type (r, V ), then there exists a

unique k-form s

on M with values in the vector bundle E = P ×

V deﬁned

as follows:

(x)(X

,...,X

)=˜uφ(u)(Y

,...,Y

), ∀x ∈ M,

where u ∈ π

−1

(x)andY

∈ T

P such that Tπ(Y

)=X

, 1 ≤ i ≤ k.Dueto

tensoriality of φ this deﬁnition of s

is independent of the choice of u and

.Wecalls

∈ Λ

(M,E)thek-form associated to φ.Wenotethatthe

one-to-one correspondence f → s

between F

(P, F )andΓ(E) extends to a

one-to-one correspondence φ → s

of tensorial forms of type (r, V )andforms

with values in the vector bundle E deﬁned above.

Given a connection 1-form ω on P we deﬁne the exterior covariant

diﬀerential

: Λ

(P, g) → Λ

k+1

(P, g)

on (k-forms on) P (with values in g)by

α(X

,...,X

)=dα(h(X

),...,h(X

)), ∀α ∈ Λ

(P, g).

We deﬁne the curvature 2-form Ω ∈ Λ

(P, g) of the connection 1-form ω

Ω := d

ω. (4.8)

It is easy to verify that Ω is a tensorial 2-form of type (ad, g)andthatit

satisﬁes the condition

dω(X, Y )=Ω(X, Y ) − [ω(X),ω(Y )]. (4.9)

Equation (4.9) is called the structure equation andisoftenwritteninthe

form

dω = Ω − ω ∧ ω.

Using deﬁnition (4.8) we obtain the Bianchi identity on the bundle P ,

Ω =0. (4.10)

The one-to-one correspondence φ → s

of tensorial forms of type (ad, g)and

forms with values in the vector bundle ad(P) deﬁned above associates to the

curvature form Ω a unique 2-form F

∈ Λ

(M,ad(P )) so that

= s

. (4.11)

The 2-form F

is called the curvature 2-form on M with values in the

adjoint bundle ad(P ) corresponding to the connection ω on P .

4.4 Connections and Curvature 125

4.4.1 Universal Connections

We now discuss an important class of principal bundles with natural connec-

tions, which play a fundamental role in the classiﬁcation of principal bundles

with connections. Let F stand for R, C,orH.OnF

we have the standard

inner product deﬁned by

u, v :=



i=1

¯u

where the bar denotes the conjugation on F , which is the identity on R and

complex (resp., quaternionic) conjugation when F is C (resp., H). Let U

(n)

denote the connected component of the identity of the Lie group of isometries

of F

(i.e., linear automorphisms of F

preserving the above inner product).

Then we have

(n)=

⎧

⎨

⎩

SO(n), if F = R,

U(n), if F = C,

Sp(n), if F = H.

The real dimension of U

(n)is

n(n +1)dim

F − n.

A k-frame in F

is an ordered orthonormal set (u

,...,u

)ofk vec-

tors in F

.Thesetofk-frames can be given a natural structure of smooth

manifold. The connected component of the k-frame (e

,...,e

), where

, 1 ≤ i ≤ n} is the set of standard orthonormal basis vectors in F

,is

called the Stiefel manifold of k-frames in F

. It is denoted by V

). The

group U

(n) acts transitively on V

). The stability group of this action

at the k-frame (e

,...,e

) can be identiﬁed with the subgroup U

(l)of

the group U

(n), where l = n − k. In block matrix form we can write an

element of the group U

(l)as



(k,n−k)

(n−k,k)



where I

is the k ×k unit matrix, O

(i,j)

is the i ×j zero matrix, and T is the

l × l matrix in U

(l) (regarded as the group of isometries of F

). Hence, we

can identify the Stiefel manifold V

)astheleftcosetspaceU

(n)/U

(l).

Thus, U

(n) is a principal U

(l)-bundle over the Stiefel manifold V

We use the following notation for indices:

1. lower case Latin letters i,j,... take values from 1 to k;

2. upper case Latin letters A,B,... take values from 1 to l = n −k;

3. Greek letters α,β,... take values from 1 to n.

Then the canonical projection π

: U

(n) → V

)isgivenby





→ (u

,...,u

) ∈ V

126 4 Bundles and Connections

where u



α=1

αj

. We note that the subgroup U

(k) of the group

(n) formed by elements of the type



(k,n−k)

(n−k,k)

n−k



acts on F

, sending a k-frame to a k-frame and inducing a free action of

(k) on the Stiefel manifold V

) on the right. The quotient manifold of

) under this action is called the Grassmann manifold and is denoted

by G

). It can be deﬁned directly as the set of k-planes through the origin

in F

. Two elements u, v ∈ V

) determine the same k-plane in F

and only if there exists t ∈ U

(k) such that ut = v.Thus,V

)canbe

regarded as a principal U

(k)-bundle over the Grassmann manifold G

)

with canonical projection π.Inparticular,G

)=CP

and each point is

a complex line. The assignment of this line to the corresponding point deﬁnes

a complex line bundle which is called the tautological complex line bundle.

Similar deﬁnition can be give for the real and quaternionic case.

Let Θ denote the canonical Cartan 1-form on the Lie group U

(n)with

values in the corresponding Lie algebra u

(n). It is often written in the form

Θ = g

−1

dg =(Θ

αβ

), (4.12)

where (Θ

αβ

)areF -valued 1-forms on U

(n) which satisfy the relations

αβ

+ Θ

αβ

=0. (4.13)

The form Θ is left invariant and right equivariant under the adjoint action

and satisﬁes the Maurer–Cartan equations

dΘ + Θ ∧ Θ =0, i.e., dΘ

αβ

+ Θ

αγ

∧ Θ

γβ

=0. (4.14)

The forms Θ

αj

are horizontal relative to the projection π

.TheformsΘ

are invariant under the action of U

(l)onU

(n) and hence project to forms

on the Stiefel manifold V

),whichwecontinuetodenotebythesame

symbols. The 1-form ω =(Θ

)withvaluesintheLiealgebrau

(k)satisﬁes

the conditions for it to be a connection form on the bundle V

) regarded

as a principal U

(k)-bundle over the Grassmann manifold G

). The con-

nection deﬁned by this form ω is called a universal connection in view of

the fact that the principal bundle V

)(G

),U

(k)) with connection

ω is m-classifying for m<kl, i.e. given a principal U

(k)-bundle P over a

compact manifold M of dim M ≤ m with connection α there exists a map

f : M → G

) such that P = f

∗

)) and α = f

∗

ω (see Chapter 5 for

further discussion).

The Stiefel and Grassmann manifolds carry a natural Riemannian struc-

ture, which can be described as follows. The quadratic form



α,j

αj

⊗Θ

αj

on U

(n) is invariant under the action of U

(l) and hence descends to the

4.5 Covariant Derivative 127

quotient to deﬁne the metric dσ

on V

) such that



α,j

αj

⊗ Θ

αj

= π

∗

(dσ

The quadratic form



A,j

⊗Θ

on U

(n) is invariant under the left U

(l)

and the right U

(k) action and therefore descends to deﬁne a Riemannian

metric ds

on G

). We have the following relations



A,j

⊗ Θ

=(π ◦ π

)

∗

(ds

), (4.15)

dσ

= π

∗

(ds



i,j

⊗ Θ

. (4.16)

We note that this metric is useful in generalized Kaluza–Klein theories. In

the special case of k =1andF = C (resp., H), the metric dσ

reduces to the

natural Riemannian metric on a sphere S

2n−1

(resp., S

4n−1

), and the metric

reduces to the Fubini–Study metric on the corresponding projective

space CP

n−1

(resp., HP

n−1

). These are the classical Hopf ﬁbrations,also

called the Hopf ﬁberings.Inthecasen = 2, the complex ﬁbration appears

in consideration of the Dirac monopole whereas the quaternionic ﬁbration

corresponds to the well-known BPST instanton solution of the Yang–Mills

equations (see Chapter 8 for further details).

We observe that U

(n)isaU

(k) ×U

(l)-bundle over G

) and hence

we may view U

(n) as a restriction of the bundle of orthonormal frames of

) corresponding to the canonical injection U

(k) × U

(l) → U

(kl).

The form δ

+ δ

on U

(n) deﬁnes the Levi-Civita connection

for G

4.5 Covariant Derivative

Let P (M,G) be a principal bundle and E(M,F,r,P) be the associated ﬁber

bundle over M with ﬁber type F and action r. A connection Γ in P allows

us to deﬁne the notion of a horizontal vector ﬁeld on E.Letw ∈ E and

(u, a) ∈O

−1

(w), where O : P × F → E is the canonical orbit projection.

Deﬁne

: P → E by u →O(u, a),

where O(u, a) is the orbit of (u, a)inE. Now deﬁne H

= H

E ⊂ T

E by

:= (f

)

∗

P ),

where H

P is the horizontal subspace of T

P .ItcanbeshownthatH

E is

independent of the choice of (u, a) ∈O

−1

(w) and is thus well-deﬁned. The