Marathe K. Topics in Physical Mathematics

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128 4 Bundles and Connections

assignment w → H

, w ∈ E, deﬁnes a connection on E. Thus, one has

the notions of horizontal vector ﬁeld on E, horizontal lift of a vector ﬁeld

on M, horizontal curve, and horizontal lift of a curve to E.Givenacurve

x :[0, 1] → M and w

∈ E such that π

)=x(0), there is a unique

horizontal lift w :[0, 1] → E of the curve x such that w(0) = w

and

(w(t)) = x(t), ∀t ∈ [0, 1],

i.e., the curve w in E is horizontal and π

(w(t)) = x(t). Given the curve x

joining x

and x

, the horizontal lift induces a diﬀeomorphism of π

−1

)and

−1

) called parallel translation or parallel displacement of ﬁbers of

E. In particular, if E is a vector bundle (i.e., the ﬁber type F is a vector

space), then parallel displacement is an isomorphism and we may deﬁne the

covariant derivative ∇

˙x(t)

s of a section s at x(t) along the vector ˙x(t)by

the formula

∇

˙x(t)

s = lim

h→0

−1

t,t+h

(s(x(t + h))) −s(x(t))],

where

t,t+h

: π

−1

(x(t)) → π

−1

(x(t + h))

is the parallel displacement along x from x(t)tox(t + h). If X ∈X(M)and

x is the integral curve of X through x

,sothatX

x(t)

=˙x(t), then the above

deﬁnition may be used to deﬁne the covariant derivative ∇

Theorem 4.5 The covariant derivative operator ∇

: Γ (E) → Γ (E)

satisﬁes the following relations, ∀X, Y ∈X(M), ∀f ∈F(M) and ∀t, s ∈

Γ (E):

1. ∇

X+Y

s = ∇

s + ∇

2. ∇

(s + t)=∇

s + ∇

3. ∇

s = f∇

4. ∇

(fs)=f ∇

s +(Xf)s.

We note that the covariant derivative may be deﬁned in terms of the Lie

derivative as follows. Recall ﬁrst that there is a one-to-one correspondence

between G-equivariant functions from P to F and sections in Γ (E), which is

deﬁned as follows. Let s ∈ Γ (E); then we deﬁne

: P → F by u → ˜u

−1

(s(x)),

where π(u)=x and ˜u : F → E

is the isomorphism deﬁned by u ∈ P.

Conversely, given a G-equivariant map f from P to F , deﬁne s

∈ Γ (E)by

(x)=O(u, f(u)),

where u ∈ π

−1

(x). This is well-deﬁned because of equivariance. The covariant

derivative ∇

s corresponds to L

)wheref

is deﬁned above and

X is

4.5 Covariant Derivative 129

the horizontal lift of X to P . The covariant derivative deﬁnes a map ∇ from

Γ (E)toΓ (T

∗

(M) ⊗ E) as follows. Recall that a section s ∈ Γ (T

∗

(M) ⊗ E)

may be regarded as deﬁning, for each x ∈ M,amaps(x):T

M → E

.For

s ∈ Γ (E), let ∇s : M → T

∗

(M) ⊗ E be the map deﬁned by

∇s(x) ·X

=(∇

s)(x).

Then we have

∇(fs)=df ⊗ s + f ∇s, ∀f ∈F(M )and∀s ∈ Γ (E). (4.17)

To indicate the dependence of ∇ on the connection 1-form ω,itiscustomary

to denote it by ∇

, and the same notation is used for its natural extension

to E-valued tensors on M . We note that the operator ∇

maps Λ

(M,E)to

(M,E). We denote the extension of the operator ∇

to Λ

(M,E)byd

in agreement with the notation used for vector-valued forms. It is deﬁned as

follows:

α(X

,...,X

):=



j=0

(−1)

∇

(α(X

,...,

,...,X

))



i<j

(−1)

i+j

α([X

],X

,...,

,...,X

where the hat sign ˆ on a vector ﬁeld denotes deletion of that vector ﬁeld.

The operator d

is called the exterior covariant diﬀerential or simply the

covariant diﬀerential on M . Applying the above equation to the curvature

2-form F

∈ Λ

(M,ad(P )) we obtain the Bianchi identity on M ,

=0. (4.18)

The classical Bianchi identity is a special case of the above identity

when ω is the Levi-Civita connection on the principal bundle of orthonormal

frames. The operator d

satisﬁes the following relation:

(β ∧ α)=(d

β) ∧ α +(−1)

deg(β)

β ∧ (d

α), (4.19)

where α ∈ Λ

∗

(M,E)and β ∈ Λ

∗

(M,R) is a homogeneous form. In fact, this

relation can be used to deﬁne the operator d

. It is customary to write d

for

the above operator if we want to emphasize its action on p-forms, otherwise

we write d

to denote any one of these operators. Thus our earlier deﬁnition

of d

: Λ

(M,E) → Λ

(M,E), combined with the above deﬁnition, gives us

the sequence

0 −→ Λ

(M,E)

−→ Λ

(M,E)

−→ Λ

(M,E)

−→··· ,

130 4 Bundles and Connections

which is called the generalized de Rham sequence.Wenotethatifα ∈

(M,E), then we have

◦ d

(α)=αΩ. (4.20)

Thus, we can consider the curvature Ω as a zeroth order operator, which acts

as an obstruction to the vanishing of (d

)

:= d

◦ d

. We will ﬁnd these

concepts useful in the study of gauge ﬁelds and their associated ﬁelds.

4.6 Linear Connections

Let L(M ) be the bundle of frames of M. Then we have seen that L(M)

is a principal bundle with structure group GL(m, R).Aconnectiononthis

principal bundle is called a linear connection.IfΓ is a linear connection

on M,the1-formω of Γ is a 1-form on L(M )withvaluesingl(m, R). Let us

denote by {u

| i, j =1, 2,...,m} the natural basis of gl(m, R), where u

the n ×n matrix such that the only non-zero element is 1 at the ith column

and jth row. Then locally we have

ω = Γ

. (4.21)

The functions Γ

are called the Christoﬀel symbols of the linear connec-

tion Γ . One can show that

∇

∂

= Γ

∂

Recall that the tangent bundle TM = E(M,R

,GL(m, R),L(M)) is a

vector bundle associated to the bundle of frames L(M). In particular, a frame

u ∈ L(M ) induces an isomorphism

˜u : R

→ T

π(u)

Let X ∈ T

L(M) and deﬁne the 1-form θ ∈ Λ

(L(M), R

)by

(X)=˜u

−1

(π

∗

(X)). (4.22)

Then θ is a tensorial 1-form on L(M )oftype(r, R

), where r is the deﬁning

representation of GL(m, R). In the physics literature the form θ is frequently

called the soldering form. If we choose a basis for the Lie algebra gl(m, R)

and a basis for R

, then we can express the connection 1-form ω as m

1-forms ω

and the form θ as m 1-forms θ

, 1 ≤ k ≤ m.Thesem

+ m

forms are globally deﬁned on L(M)andmakeL(M )intoaparallelizable

manifold. In particular, we can deﬁne a Riemannian metric on L(M)by

4.6 Linear Connections 131



i=1



j=1



k=1

. (4.23)

Using this Riemannian metric we can make L(M) into a metric space. This

metric can be used to show that a manifold M that admits a linear connection

(in particular, a Lorentz connection) must be a metric space. Thus, in consid-

ering manifolds of interest in physical applications one may restrict attention

to manifolds that are metric spaces. The metric on L(M) is called the bun-

dle metric or b-metric andisusedindeﬁningtheso-calledb-boundaries

for space-time manifolds. The bundle metric was introduced by Marathe in

[265] to study the topology of spaces admitting a linear connection and in-

dependently by Schmidt in [338] in his study of singular points in general

relativity.

We observe that the linear connections are distinguished from connections

in other principal bundles by the existence of the soldering form. Thus, in

addition to the curvature 2-form Ω = d

ω,wehavethetorsion 2-form

ϑ = d

θ ∈ Λ

(L(M), R

). A linear connection ω is called torsion free if

the torsion 2-form ϑ = d

θ =0.

Let ω be a linear connection on the m-manifold M, i.e., a connection on

the principal GL(m, R)-bundle of frames of M.Ifx ∈ M, X

∈ T

M,let

us consider the element

˜u(ϑ(X

∗

)) ∈ T

where u ∈ L(M )

and X

∗

are elements of T

L(M) such that π

∗

,π

∗

= Y

. One can show that ˜u(ϑ(X

∗

)) does not depend on the

choices of u, X

∗

. Thus, it is easy to see that

T (X

):=˜u(ϑ(X

∗

)),X

∈ T

deﬁnes a tensor ﬁeld T ∈ Γ (T

(M)), which is called the torsion tensor

ﬁeld, or simply the torsion. Analogously, the curvature tensor ﬁeld or

simply the curvature is the tensor ﬁeld R ∈ Γ (T

(M)) such that

R(X

:= ˜u[Ω(X

∗

)(˜u

−1

))],

∀x ∈ M, X

∈ T

M.WeobservethatΩ(X

∗

)isintheLiealgebra

gl(m, R)ofGL(m, R), while ˜u

−1

) ∈ R

and thus Ω(X

∗

)(˜u

−1

)) ∈

. One can show that the torsion T and the curvature R can be expressed

in terms of the covariant derivative ∇≡∇

as follows:

T (X, Y )=∇

Y −∇

X − [X, Y ], (4.24)

R(X, Y )Z =[∇

, ∇

]Z −∇

[X,Y ]

Z. (4.25)

Locally, with X = X

∂

,Y= Y

∂

132 4 Bundles and Connections

T (X, Y )=(Γ

− Γ

∂

(4.26)

and thus

≡ dx

(T (∂

,∂

)) = Γ

− Γ

. (4.27)

From the above local expression it follows that a linear connection is torsion-

free if and only if its Christoﬀel symbols are symmetric in the lower indices.

Hence, a torsion-free connection is sometimes called a symmetric connec-

tion. Analogously, deﬁning

ijk

:= dx

(R(∂

,∂

)∂

), (4.28)

we have

ijk

= ∂

− ∂

+ Γ

− Γ

. (4.29)

We observe that

ijk

= −R

ikj

(4.30)

and, for a torsion-free connection,

[ijk]

≡ R

ijk

+ R

kij

+ R

jki

=0. (4.31)

By contraction, from the curvature one obtains the following important tensor

ﬁeld. The Ricci tensor ﬁeld is the tensor ﬁeld Ric ∈ Γ (T

(M)) deﬁned by

Ric(X

)=α

(R(e

∀x ∈ M, X

∈ T

M,where{e

,...,e

} is a basis of T

M with dual

basis {α

,...,α

}.Thuslocally,thecomponentsR

of the Ricci tensor

are given by

:= Ric(∂

,∂

)=R

jki

. (4.32)

If M is a Riemannian manifold, then the structure grou GL(m, R)can

be reduced to the orthogonal group O(m, R). A connection on the reduced

bundle is called a Riemannian connection. Similarly if M is a Lorentz

manifold, then the structure group GL(m, R) can be reduced to the Lorentz

group. The frames in the reduced bundle are called local inertial frames

and the connection on the reduced bundle is called a Lorentz connection.

If m = 4 then the Lorentz manifold M is called a space-time manifold.The

connection and curvature can be interpreted as representing a gravitational

potential and gravitational ﬁeld when they satisfy gravitational ﬁeld

equations. We will discuss them in Chapter 6. The Levi-Civita connection

deﬁned in the next paragraph is determined by the metric tensor g and hence

in this case g is interpreted as representing the gravitational potential.

Einstein’s ﬁeld equations are then expressed as non-linear partial diferential

equations for the components of g.

A linear connection ω on a pseudo-Riemannian manifold (M, g) is called a

metric connection if the metric g is covariantly constant, i.e., ∇

g =0.

4.6 Linear Connections 133

Among all metric connections on a pseudo-Riemannian manifold there ex-

ists a unique torsion-free, metric connection λ called the Levi-Civita con-

nection. This fact is sometimes referred to as the fundamental theorem of

pseudo-Riemannian geometry. The Levi-Civita connection is also referred to

as the symmetric connection. The curvature 2-form F

of the Levi-Civita

connection λ is denoted by R and is called the Riemann curvature of M.

Locally, the Christoﬀel symbols of the Levi-Civita connection are given,

in terms of the metric g,by

(∂

+ ∂

− ∂

). (4.33)

The Ricci tensor of a pseudo-Riemannian manifold (M,g) is symmetric. Other

important quantities for (M,g) are the scalar curvature and the sectional

curvature which we now deﬁne. The scalar curvature is the function S ∈

F(M) deﬁned by

S = g

. (4.34)

Using this deﬁnition we obtain the following expression for the trace-free

part K of the Ricci tensor

= R

−

. (4.35)

Let p be a 2-dimensional subspace of T

M and let {e

} be an orthonormal

basis for p.Thequantity

κ(p):=e



(R(e

) (4.36)

does not depend on the choice of the orthonormal basis {e

} and is called

the sectional curvature along p.Thelocalexpressionforthecovariant

derivative

∇

Y =(δ

+ Γ

)δ

,X,Y∈X(M) (4.37)

shows that (∇

Y )(x) depends on the value of X only in x. This allows us to

deﬁne the covariant derivative of a vector ﬁeld X along a curve c : I → M as

the curve

: I → TM

in TM given by

(t)=(∇

˙c

X)(c(t)),

where ˙c(t)=Tc(t, 1). The vector ﬁeld X is said to be autoparallel along c

if ∇

˙c

X =0.Ageodesic is a curve γ in M that is autoparallel along itself,

i.e., it satisﬁes the equation

∇

˙γ

˙γ =0. (4.38)

In local coordinates this gives the system of diﬀerential equations

134 4 Bundles and Connections

¨γ

+ Γ

˙γ

=0, (4.39)

where i, j, k go from 1 to dim M and the Einstein summation convention is

used. These equations always admit a local solution giving a geodesic starting

at a given point in a given direction. On a complete manifold there exists a

piecewise smooth geodesic joining any two points. The classical Hopf–Rinow

theorem states that on a complete manifold every geodesic can be extended to

a geodesic deﬁned for all time t ∈ R. In classical mechanics a particle of unit

mass moving with velocity v has kinetic energy

.Foracurvec : I → M

the tangent vector ˙c(t) is the velocity vector at time t.LetP

denote the set

of all smooth curves from I to M . Then it can be shown that the critical

points of the energy functional

E(c)=



g(˙c, ˙c)dt

when c varies over P

are the geodesics.

Let M be an oriented Riemannian manifold. The identiﬁcation of the Lie

algebra so(m)withΛ

) allows us to identify ad L(M)withΛ

(M). Thus

for each x ∈ M, R deﬁnes a symmetric, linear transformation of Λ

(M). The

dimension 4 is further distinguished by the fact that so(4) = so(3) ⊕ so(3)

and that this decomposition corresponds to the decomposition Λ

(M)=

(M) ⊕ Λ

−

(M)into±1 eigenspaces of the Hodge star operator. The Rie-

mann curvature R also decomposes into SO(4)-invariant components induced

by the above direct sum decomposition of Λ

(M). These components are the

self-dual (resp., anti-dual) Weyl tensor W

(resp., W

−

), the trace-free

part K of the Ricci tensor, and the scalar curvature S.Thus

R = W

⊕W

−

⊕ (K ×

g) ⊕ (g ×

g)S (4.40)

where ×

is the curvature product deﬁnedinSection6.7. These compo-

nents can be used to deﬁne several important classes of 4-manifolds. Thus,

M is called a self-dual manifold (resp., anti-dual manifold)ifW

−

(resp., W

= 0 ). It is called conformally ﬂat if it is both self-dual and

anti-dual or if the full Weyl tensor is zero, i.e., W := W

+ W

−

=0.A

pseudo-Riemannian manifold (M,g)issaidtobeanEinstein manifold if

Ric = Λg , (4.41)

where Λ is a constant. By contraction of both sides of equation (4.41)we

get Λ = S/m. Hence equation (4.41) is equivalent to the vanishing of the

trace-free part K of the Ricci tensor, i.e., K = 0. Einstein manifolds corre-

spond to a class of gravitational instantons ([269]). Einstein manifolds were

characterized in [353] by the commutation condition [R, ∗]=0,wherethe

Riemann curvature R and the Hodge star operator ∗ are both regarded as

linear transformations of Λ

(M). This condition was generalized in [268,267]

4.7 Generalized Connections 135

to obtain a new formulation of the gravitational ﬁeld equations, which are

discussed in in Section 6.7. For the study of Riemannian manifolds in dimen-

sion 4 and manifolds of diﬀerentiable mappings see, for example, Donaldson

and Kronheimer [112].

4.7 Generalized Connections

The various deﬁnitions of connection in principal and associated bundles

discussed in this chapter are adequate for most of the applications discussed

in this book. However, it is possible todeﬁnethenotionofconnectiononan

arbitrary ﬁber bundle. We call this a generalized connection. It can be used

to give an alternative formulation of some aspects of gauge theories. There

is extensive work in this area but we will not use this approach in our work.

Let E be a ﬁber bundle over B with projection p : E → B.LetVEdenote

the vertical vector bundle over E. VE is a subbundle of the tangent bundle

TE, the ﬁber V

E, u ∈ E being the tangent space to the ﬁber E

p(u)

of E

passing through u.Letp

∗

(TB) be the pull-back of the tangent bundle of B

to E. Then we have the following short exact sequence of vector bundles and

morphisms over E:

0 −→ VE

−→ TE

−→ π

∗

(TB) −→ 0 , (4.42)

where i is the injection of the vertical bundle VEinto the tangent bundle, and

a is deﬁned by (e, X

) → (e, p

∗

)). We deﬁne a generalized connection

on E to be a splitting of the above exact sequence, i.e., a vector bundle

morphism c : π

∗

(TB) → TE such that a ◦ c = id

∗

(TB)

.Letb ∈ B and let

e ∈ p

−1

(b), then the splitting c induces an injection ˆc

: T

(B) → T

(E),X→

c(e, X). We call ˆc

(X)thehorizontal lift of the tangent vector X to e ∈ E.

We call ˆc

(B)) the horizontal space at e ∈ E and denote it by H

The spaces H

E are the ﬁbers of the horizontal bundle HE and we have the

decomposition

TE = HE ⊕ VE.

We note that this decomposition corresponds to the ﬁrst condition of Def-

inition 4.5. Alternatively, a connection may be deﬁned as a section of the

ﬁrst jet bundle J

(E)overE. The deﬁnitions of covariant derivative, covari-

ant diﬀerential, and curvature can be formulated in this general context. An

introduction to this approach and to its physical applications is given in [153].

If E = P (M,G), then we can recover the usual deﬁnition of connection as

follows. The action of G on P extends to all the vector bundles in the exact

sequence (4.42) to give the following short exact sequence of vector bundles

over M:

0 −→ VP/G

−→ TP/G

ˆa

−→ π

∗

(TM)/G −→ 0.

136 4 Bundles and Connections

We can rewrite the above sequence as follows:

0 −→ ad(P )

−→ TP/G

ˆa

−→ TM −→ 0. (4.43)

A connection on P is then deﬁned as a splitting of the short exact se-

quence (4.43). This splitting induces a splitting of the sequence (4.42)and

we recover Deﬁnition 4.5 of the connection given earlier. This approach helps

to clarify the role of the structure group in the usual deﬁnition of connection

in a principal bundle.

Chapter 5

Characteristic Classes

5.1 Introduction

In 1827 Gauss published his classic book Disquisitiones generales circa su-

perﬁcies curvas. He deﬁned the total curvature (now called the Gaussian

curvature) κ as a function on the surface. In his famous theorema egregium

Gauss proved that the total curvature κ of a surface S depends only on the

ﬁrst fundamental form (i.e., the metric) of S. Gauss deﬁned the integral

curvature κ(Σ) of a bounded surface Σ to be



κdσ. He computed κ(Σ)

when Σ is a geodesic triangle to prove his celebrated theorem

κ(Σ):=



κdσ= A + B + C −π, (5.1)

where A, B, C are the angles of the geodesic triangle Σ. Gauss was aware

of the signiﬁcance of equation (5.1) in the investigation of the Euclidean

parallel postulate (see Appendix B for more information). He was inter-

ested in surfaces of constant curvature and mentions a surface of revolution

of constant negative curvature, namely, a pseudosphere. The geometry of

the pseudosphere turns out to be the non-Euclidean geometry of Lobaˇcevski–

Bolyai.

Equation (5.1) is a special case of the well-known Gauss–Bonnet the-

orem. When applied to a compact, connected surface Σ the Gauss-Bonnet

theorem states that



κdσ=2πχ(Σ), (5.2)

where χ(Σ) is the Euler characteristic of Σ. The left hand side of equation

(5.2) is arrived at through using the diﬀerential structure of Σ, while the

right hand side depends only on the topology of Σ. Thus, equation (5.2) is

a relation between geometric (or analytic) and topological invariants of Σ.

This result admits far-reaching generalizations. In particular, it can be re-

garded as a prototype of an index theorem. The Gauss–Bonnet theorem was

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 5, 137

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