Gauge mechanics-Sardanashvily

8/7/2019 Gauge mechanics-Sardanashvily

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Gauge Mechanics(World Scientific, Singapore, 1998 )

L. MANGIAROTTI, G. SARDANASHVILY


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Preface

This book presents in a unified way modern geometric methods in analytical me-chanics, based on the application of jet manifolds and connections. As is well known,

the technique of Poisson and symplectic spaces provide the adequate Hamiltonian

formulation of conservative mechanics. This formulation, however, cannot be ex-

tended to time-dependent mechanics subject to time-dependent transformations.

We will formulate non-relativistic time-dependent mechanics as a particular field

theory on fibre bundles over a time axis.

The geometric approach to field theory is based on the identification of classical

fields with sections of fibred manifolds. Jet manifolds provide the adequate mathe-

matical language for Lagrangian field theory, while the Hamiltonian one is phrasedin terms of a polysymplectic structure. The 1-dimensional reduction of Lagrangian

field theory leads us in a straightforward manner to Lagrangian time-dependent

mechanics. At the same time, the canonical polysymplectic form on a momentum

phase space of time-dependent mechanics reduces to the canonical exterior 3-form

which plays the role similar to a symplectic form in conservative mechanics. With

this canonical 3-form, we introduce the canonical Poisson structure and formulate

Hamiltonian time-dependent mechanics in terms of Hamiltonian connections and

Hamiltonian forms.

Note that the theory of non-linear differential operators and the calculus of vari-

ations are conventionally phrased in terms of jet manifolds. On the other hand, jet

formalism provides the contemporary language of differential geometry to deal with

non-linear connections, represented by sections of jet bundles. Only jet spaces enable

us to treat connections, Lagrangian and Hamiltonian dynamics simultaneously.

In fact, the concept of connection is the main link throughout the book. Con-

nections on a configuration space of time-dependent mechanics are reference frames.

Holonomic connections on a velocity phase space define non-relativistic dynamic

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vi

equations which are also related to other types of connections, and can be writ-ten as non-relativistic geodesic equations. Hamiltonian time-dependent mechanics

deals with Hamiltonian connections whose geodesics are solutions of the Hamilton

equations.

The presence of a reference frame, expressed in terms of connections, is the main

peculiarity of time-dependent mechanics. In particular, each reference frame defines

an energy function, and quantizations with respect to different reference frames are

not equivalent.

Another important peculiarity is that a Hamiltonian fails to be a scalar function

under time-dependent transformations. As a consequence, many well-known con-structions of conservative mechanics fail to be valid for time-dependent mechanics,

and one should follow methods of field theory.

At the same time, there is the essential difference between field theory and time-

dependent mechanics. In contrast with gauge potentials in field theory, connections

on a configuration space of time-dependent mechanics fail to be dynamic variables

since their curvature vanishes identically. Following geometric methods of field the-

ory, we obtain the frame-covariant formulation of time-dependent mechanics. By

analogy with gauge field theory, one may speak about gauge time-dependent me-

chanics.

In comparison with non-relativistic time-dependent mechanics, a configuration

space of relativistic mechanics does not imply any preferable fibration over a time.

To construct the velocity phase space of relativistic mechanics, we therefore use

formalism of jets of submanifolds. At the same time, Hamiltonian relativistic me-

chanics is seen as an autonomous Hamiltonian system on the constraint space of

relativistic hyperboloids.

With respect to mathematical prerequisites, the reader is expected to be familiar

with the basics of differential geometry of fibre bundles. For the convenience of the

reader, several mathematical facts and notions are included as an Interlude, thus

making our exposition self-contained.


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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Interlude: bundles, jets, connections 9

1.1 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Multivector fields and differential forms . . . . . . . . . . . . . . . . . 20

1.3 Jet manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.4 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.5 Bundles with symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.6 Composite fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . 53

2 Geometry of Poisson manifolds 59

2.1 Jacobi structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2 Contact structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.3 Poisson structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.4 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.5 Presymplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.6 Reduction of symplectic and Poisson structures . . . . . . . . . . . . 86

2.7 Appendix. Poisson homology and cohomology . . . . . . . . . . . . . 91

2.8 Appendix. More brackets . . . . . . . . . . . . . . . . . . . . . . . . . 982.9 Appendix. Multisymplectic structures . . . . . . . . . . . . . . . . . . 103

3 Hamiltonian systems 107

3.1 Dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.2 Poisson Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . 113

3.3 Symplectic Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . 116

3.4 Presymplectic Hamiltonian systems . . . . . . . . . . . . . . . . . . . 121

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viii CONTENTS

3.5 Dirac Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . 1263.6 Dirac constraint systems . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.7 Hamiltonian systems with symmetries . . . . . . . . . . . . . . . . . . 136

3.8 Appendix. Hamiltonian field theory . . . . . . . . . . . . . . . . . . . 142

4 Lagrangian time-dependent mechanics 153

4.1 Fibre bundles over R . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.2 Dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.3 Dynamic connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.4 Non-relativistic geodesic equations . . . . . . . . . . . . . . . . . . . 1724.5 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.6 Free motion equations . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.7 Relative acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.8 Lagrangian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.9 Newtonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

4.10 Holonomic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 209

4.11 Non-holonomic constraints . . . . . . . . . . . . . . . . . . . . . . . . 214

4.12 Lagrangian conservation laws . . . . . . . . . . . . . . . . . . . . . . 221

5 Hamiltonian time-dependent mechanics 2295.1 Canonical Poisson structure . . . . . . . . . . . . . . . . . . . . . . . 230

5.2 Hamiltonian connections and Hamiltonian forms . . . . . . . . . . . . 233

5.3 Canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . 244

5.4 The evolution equation . . . . . . . . . . . . . . . . . . . . . . . . . . 249

5.5 Degenerate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

5.6 Quadratic degenerate systems . . . . . . . . . . . . . . . . . . . . . . 264

5.7 Hamiltonian conservation laws . . . . . . . . . . . . . . . . . . . . . . 271

5.8 Time-dependent systems with symmetries . . . . . . . . . . . . . . . 273

5.9 Systems with time-dependent parameters . . . . . . . . . . . . . . . . 2765.10 Unified Lagrangian and Hamiltonian formalism . . . . . . . . . . . . 285

5.11 Vertical extension of Hamiltonian formalism . . . . . . . . . . . . . . 288

5.12 Appendix. Time-reparametrized mechanics . . . . . . . . . . . . . . . 298

6 Relativistic mechanics 301

6.1 Jets of submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

6.2 Relativistic velocity and momentum phase spaces . . . . . . . . . . . 305


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CONTENTS ix

6.3 Relativistic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3096.4 Relativistic geodesic equations . . . . . . . . . . . . . . . . . . . . . . 313

7 Appendix A. Geometry of BRST mechanics 319

8 Appendix B. On quantum time-dependent mechanics 329

Bibliography 333


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x CONTENTS


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Introduction

The present book deals with first order mechanical systems, governed by the second

order differential equations in coordinates or the first order ones in coordinates

and momenta. Our goal is the description of non-conservative mechanical systems

subject to time-dependent transformations, including inertial and non-inertial frame

transformations and phase transformations.

Symplectic technique is well known to provide the adequate Hamiltonian for-

mulation of conservative (i.e., time-independent) mechanics where Hamiltonians are

independent of time [2, 6, 72, 116, 126]. The familiar example is a mechanical sys-

tem whose momentum phase space is the cotangent bundle TM of a configuration

space M. This fibre bundle is provided with the canonical symplectic form

= dpi dqi, (0.0.1)

written with respect to the holonomic coordinates (qi, pi = qi) on TM. A Hamil-

tonian H of a conservative mechanical system is defined as a real function on themomentum phase space TM. Then a motion of this system is an integral curve of

the Hamiltonian vector field

= ii + ii

on TM which fulfills the Hamilton equations

= dH,i = iH, i = iH.

Lagrangian conservative mechanics is usually seen as a particular Hamiltonian me-

chanics on the tangent bundle T M of a configuration space M, which is endowed

with the presymplectic form defined by a Lagrangian.

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2 INTRODUCTION

The Hamiltonian formulation of conservative mechanics cannot be extended in astraightforward manner to time-dependent mechanics because the symplectic form

(0.0.1) is not invariant under time-dependent transformations, including the inertial

frame transformations.

The existent formulation of time-dependent mechanics implies a preliminary

splitting of a configuration space

Q = R M, (0.0.2)where M is a manifold, while R is a time axis (see [29, 31, 50, 110, 136, 146, 166] and

references therein). From the physical viewpoint, it means that a certain referenceframe is chosen. Then we have the corresponding splitting of the velocity phase

space

R T M (0.0.3)and that of the momentum phase space

R TM. (0.0.4)The momentum phase space (0.0.4) is provided with the presymplectic form

pr2 = dpi dqi (0.0.5)which is the pull-back of the canonical symplectic form (0.0.1) on the cotangent

bundle TM [27]. By a time-dependent Hamiltonian H is meant a real functionon the momentum phase space R TM, while trajectories of motion are integralcurves of the time-dependent vector field

: R TM T TMwhich satisfies the Hamilton equations

i = iH, i = iH.The problem is that the splittings (0.0.2) (0.0.4) are broken by any time-

dependent transformation, and so is the presymplectic form (0.0.5). Therefore the

familiar methods of conservative mechanics and their extensions to the product

spaces (0.0.2) (0.0.4) fail to be valid for mechanical systems subject to time-

dependent transformations.


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INTRODUCTION 3

We will formulate non-relativistic time-dependent mechanics as a particular fieldtheory whose configuration space is a fibred manifold over a time axis R [14, 55, 57,

106, 114, 132, 159, 161].

Geometric formalism of field theory is based on the identification of classical

fields with sections of a fibred manifold Y X. The corresponding velocity phasespace is the first order jet manifold J1Y of sections ofY X, while the momentumone is the Legendre bundle

= VY (n1 TX), n = dim X, (0.0.6)

over Y [28, 56, 57, 73, 96, 158, 159].In the case of X = R of time-dependent mechanics, its configuration space is a

fibred manifold

: Q R,equipped with fibred coordinates (t, qi). The base R is parameterized by the Carte-

sian coordinates t with the transition functions t = t+const. Relative to these

coordinates, the time axis R is provided with the standard vector field t and the

standard 1-form dt which is also the volume element on R. Of course, this is not

the case of relativistic mechanics (see Chapter 6) nor of the models with a time

reparametrization (see Section 5.12).

The velocity phase spaceof non-relativistic time-dependent mechanics is the first

order jet manifold J1Q of sections of the fibred manifold Q R. It is equippedwith the adapted coordinates (t, qi, qit). There is the canonical imbedding

: J1Q Q

T Q, (0.0.7)

= t + qiti,

of the velocity phase space J1Q into the tangent bundle T Q of the configuration

space Q. From now on we will identify J

1

Q with its image in T Q given by thecoordinate conditions

t = 1, qi = qit. (0.0.8)

This is an affine subbundle of T Q Q modelled over the vertical tangent bundleV Q Q of the fibred manifold Q R.

The morphism (0.0.7) plays a prominent role in our formulation of time-dependent

mechanics. It enables us to treat the jet manifold J1Q as a velocity phase space


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4 INTRODUCTION

of a mechanical system. Due to this morphism, every connection on the fibredmanifold Q R can be identified with the nowhere vanishing vector field

: Q J1Q T Q, (0.0.9) = t +

ii,

on Q which is the horizontal lift of the standard vector field t on R by means of

this connection . We will continue to call (0.0.9) a connection in order to refer

to the standard properties of connections without additional explanation. From

the physical viewpoint, a connection (0.0.9) sets a tangent vector at each pointof the configuration space Q, which characterizes the velocity of an observer at

this point. It follows that a connection on the fibred manifold Q R defines areference frame [57, 132, 161]. In particular, one can think of the difference qit ias being the relative velocity with respect to the reference frame , whereas the

notion of a relative acceleration is more intricate (see Section 4.7).

The momentum phase space of non-relativistic time-dependent mechanics is the

Legendre bundle (0.0.6) where X = R. This phase space is isomorphic to the vertical

cotangent bundle = VQ of the fibred manifold Q R, and is equipped with theholonomic coordinates (t, qi, pi = qi). It should be emphasized that this is not the

most general case of a momentum phase space of time-dependent mechanics, which

is defined as a fibred manifold R provided with a Poisson structure such thatthe corresponding symplectic foliation belongs to the fibration R [74]. In fact,putting = VQ, we restrict our consideration to Hamiltonian systems which have

the Lagrangian counterparts.

Note that Lagrangian and Hamiltonian formalisms are equivalent only if a La-

grangian is hyperregular, i.e., the Legendre map from the velocity phase space to the

momentum one is a diffeomorphism. In general, a degenerate Lagrangian involves a

set of associated Hamiltonians in order to exhaust solutions of the Lagrange equa-

tion (see Section 5.5). Nevertheless, there are physically interesting systems whosephase spaces fail to be the cotangent bundles of configuration spaces, and they do

not admit any Lagrangian description [168]. The unified LagrangianHamiltonian

formalism of the joint velocity-momentum phase space

= VJ1Q = J1VQ

enables us to relate a Lagrangian system to any Hamiltonian one (see Section 5.10).


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INTRODUCTION 5

Let us turn to the momentum phase space VQ of time-dependent mechanics.It is endowed with the canonical exterior 3-form

= dpi dqi dt (0.0.10)which is the particular case of the canonical polysymplectic form on the Legendre

bundle (0.0.6), when X = R [57, 161]. The exterior form (0.0.10) is invariant under

all holonomic transformations of the momentum phase space VQ.

In time-dependent mechanics, the canonical 3-form (0.0.10) plays a role sim-

ilar to the canonical symplectic form (0.0.1) in conservative symplectic mechanics.

The form (0.0.10) yields the canonical Poisson structure on the momentum phasespace VQ, and provides the Hamiltonian formulation of time-dependent mechanics

in terms of Hamiltonian connections and Hamiltonian forms. This formulation is

compatible with the Lagrangian formulation of time-dependent mechanics on the

velocity phase space J1Q, and is equivalent to the Lagrangian one in the case of

hyperregular Lagrangians.

The following peculiarities of Hamiltonian time-dependent mechanics should be

emphasized.

The canonical Poisson structure defined by the 3-form (0.0.10) on the mo-

mentum phase space VQ of time-dependent mechanics is degenerate.

A Hamiltonian on a momentum phase space of time-dependent mechanics failsto be a scalar function, but reads

H = pii + H, (0.0.11)where is a connection on the fibred manifold Q R, while H is a Hamil-tonian function which is also an energy density with respect to the reference

frame .

A Poisson bracket of a Hamiltonian (0.0.11) with functions on a momentumphase space is defined only locally. Being equal to zero with respect to some

coordinates, it does not necessarily vanish with respect to other ones.

As a consequence, the evolution equation in time-dependent mechanics is notreduced to a Poisson bracket, and integrals of motion are not functions in invo-

lution with a Hamiltonian. For the same reason, the familiar DiracBergmann


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INTRODUCTION 7

Therefore, Lagrangians in time-dependent mechanics are covariant, but not invariantunder reference frame transformations.

Connections play a prominent role in our formulation of time-dependent mechan-

ics. As was mentioned above, connections on a configuration bundle Q R describenon-relativistic reference frames. Holonomic connections on the jet bundle J1Q Rdefine non-relativistic dynamic equation which, in turn, are associated with connec-

tions on the affine jet bundle J1Q Q and the tangent bundle T Q Q. A s aresult, every non-relativistic dynamic equation can be seen as a geodesic equation on

the tangent bundle T Q Q that furnishes the relationship between non-relativisticand relativistic dynamics (see Section 6.4). Hamiltonian time-dependent mechanicsdeals with Hamiltonian connections whose geodesics are solutions of the Hamilton

equations.

In comparison with non-relativistic mechanics, if a configuration space of a me-

chanical system has no preferable fibration Q R, we obtain the general formula-tion of relativistic mechanics, including Special Relativity on the Minkowski space

Q = R4. The velocity phase space of relativistic mechanics is the first order jet

manifold J11Q of 1-dimensional submanifolds of the configuration space Q [57, 161].

This notion of jets generalizes that of jets of sections of fibre bundles which we have

utilized in field theory and non-relativistic mechanics. The jet bundle J11Q

Q is

projective, and one can think of its fibres as being spaces of the 3-velocities of a rela-

tivistic system. The 4-velocities of a relativistic system are represented by elements

of the tangent bundle T Q of the configuration space Q, while the cotangent bundle

TQ, endowed with the canonical symplectic form, plays the role of the momentum

phase space of relativistic theory. As a result, Hamiltonian relativistic mechanics

can be seen as a constraint Dirac system on the hyperboloids of relativistic momenta

in the momentum phase space TQ.

Formalism of jets of submanifolds provides the common description of non-

relativistic mechanics and relativistic theory. In particular, the tangent bundle T Q

of a configuration space Q plays the role of the space of the 4-velocities both in non-relativistic and relativistic mechanics. The difference is only that, given a fibration

Q R, the 4-velocities of a non-relativistic system live in the subbundle (0.0.8) ofT Q, whereas the 4-velocities of a relativistic theory belong to the hyperboloids

gqq = 1, (0.0.13)

where g is an admissible pseudo-Riemannian metric in T Q. Moreover, as was men-

tioned above, both relativistic and non-relativistic equations of motion can be seen


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8 INTRODUCTION

as geodesic equations on the tangent bundle T Q, but their solutions live in its dif-ferent subbundles (0.0.13) and (0.0.8).

Unless otherwise stated, we believe that all quantities are physically dimension-

less. Following field theory, we will sometimes refer to the universal unit system

where the velocity of light c and the Planck constant h are equal to 1, while the

length unit is the Planck one

(Ghc3)1/2 = G1/2 = 1, 616 1033cm,where G is the Newtonian gravitational constant. Relative to the universal unit

system, the physical dimension of the spatial and temporal Cartesian coordinatesis the [length], the physical dimension of a mass is the [length]1, while an action

functional and a metric tensor are physically dimensionless.


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Chapter 1

Interlude: bundles, jets,

connections

This Chapter does not claim to be a survey on modern differential geometry. The

relevant material is presented in a fairly informal way. For details, we refer the

reader to [57, 100, 157, 164, 170, 185].

Throughout the book, all maps are smooth, i.e., of class C, while manifolds

are real, finite-dimensional, second-countable and, hence, paracompact. Unless oth-

erwise stated, we assume that manifolds are connected.

We use the standard symbols , , and for the tensor, symmetric, and exteriorproducts, respectively. The interior product (contraction) of vectors and forms is

denoted by . By AB are meant the partial derivatives with respect to the coordinateswith indices AB. The symbol stands for a composition of maps.

1.1 Fibre bundles

Subsections: Fibre bundles, 9; Vector bundles, 12; Affine bundles, 14; Tangent and

cotangent bundles, 15; Tangent and cotangent bundles of fibre bundles, 16; Sheaves,

18.

Fibre bundles

By a fibre bundle is meant a locally trivial fibred manifold

: Y X (1.1.1)

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10 CHAPTER 1. INTERLUDE: BUNDLES, JETS, CONNECTIONS

where a fibration (or a projection) is a surjective submersion from a manifold Y,called a total space, onto a baseX. Unless otherwise stated, we put dim X = n. By

definition, a base X of a fibre bundle (1.1.1) admits an open covering {U} so thatY is locally isomorphic to the splittings

: 1(U) U V,

called local bundle trivializations, together with the transition functions

: (U U) V (U U) V,

(y) = ( )(y), y 1

(U U),where V is the typical fibre of the fibre bundle (1.1.1). The bundle trivializations

(U, ) constitute an atlas

= {(U, ), }of a fibre bundle. Given an atlas , a fibre bundle Y is provided with the associated

atlas of fibred coordinates (x, yi), where

x(y) = (x )(y), y Y,are coordinates on the base X, and

yi(y) = (yi pr2 )(y)are coordinates on the typical fibre V.

A fibre bundle Y X is called trivial ifY is diffeomorphic to the product XV.Different trivializations of a fibre bundle differ from each other in projections Y Vof the total space Y onto the typical fibre V.

Theorem 1.1.1. [170]. Each fibre bundle over a contractible base is trivial. 2

By a section (or a global section) of a fibre bundle (1.1.1) is meant a manifold

morphism s : X Y such that s = Id X. A section s is an imbedding, i.e.,s(X) Y is a submanifold of a total space Y which is also a topological subspaceof Y. Similarly, a section s of a fibre bundle Y X over a submanifold N X isa morphism s : N Y, such that

s = iN : N Y.


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1.1. FIBRE BUNDLES 11

A section of a fibre bundle over an open subset of its base will be called simply a(local) section. A fibre bundle, by definition, admits a local section over an open

neighbourhood of each point of its base.

Theorem 1.1.2. [170]. A fibre bundle Y X whose typical fibre is diffeomorphicto Rm has a global section. A (smooth) section over a closed subset ofX can always

be extended to a global section. 2

A fibred morphism of two bundles : Y X and : Y X is a pair of maps : Y

Y and f : X

X such that the diagram

Y Y

? ?X

fX

(1.1.2)

is commutative, i.e., sends fibres to fibres. In brief, we will say that (1.1.2) is a

fibred morphism

: Y f

Y

over f. If f = Id X, then

: Y X

Y

is called a fibred morphism over X.

Remark 1.1.1. Unless otherwise stated, by the rank of a fibred morphism (1.1.2)

over a diffeomorphism f is meant its rank minus dim X. A fibred morphism (1.1.2) over X (or its image (Y)) is said to be a subbundle

of the fibre bundle Y

X if (Y) is a submanifold of Y.

We deduce from the implicit function theorem the following useful criteria foran image and a pre-image of a fibred morphism to be a subbundle [149, 185].

Theorem 1.1.3. Let : Y Y be a fibred morphism over X. Given a globalsection s of the fibre bundle Y X such that s(X) Im , by the kernel of thefibred morphism with respect to the section s is meant the pre-image

Ker s = 1(s(X))


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of s(X) by . If : Y Y is a fibred morphism of constant rank over X, thenIm and Ker s are subbundles of Y

and Y, respectively. 2

An isomorphism of fibre bundles is a fibred morphism (1.1.2) such that is a

diffeomorphism. A fibred morphism [isomorphism] of a fibre bundle Y X to itselfis called an endomorphism [automorphism]. An automorphism over Id X is said to

be a vertical automorphism. Following physical terminology, automorphisms of fibre

bundles will also be called gauge transformations.

Given a fibre bundle : Y X and a manifold map f : X X, the pull-back

f

Y of Y by f is the fibre bundle over X

whose total space is

fYdef={(x, y) X Y : (y) = f(x)}

together with the natural projection (x, y) x. Roughly speaking, the fibre ofthe pull-back fY over a point x X is that of Y over the point f(x) X. IfX X is a submanifold of X and iX is the corresponding natural injection, thenthe pull-back

iXY = Y |X

is called the restriction of a fibre bundle Y to the submanifold X X.Let : Y X and : Y X be fibre bundles over the same base X. Their

fibred product

Y Y

over X is defined as the pull-back

Y X

Y = Y or Y X

Y = Y

together with the natural projection onto X.

Vector bundles

A vector bundleis a fibre bundle Y X such that:

its typical fibre V and all the fibres Yx = 1(x), x X, are real finite-dimensional vector spaces;


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there is a bundle atlas = {(U, )} of Y X whose trivialization mor-phisms restrict to linear isomorphisms

(x) : Yx V, x U.

Dealing with a vector bundle Y, we always use linear bundle coordinates (x, yi)

associated with the above-mentioned bundle atlas . We have

(pr2 )(y) = yiei,y = y

i

ei(x) = yi

(x)1

(ei),

where {ei} is a fixed basis for the typical fibre V of Y, while {ei(x)} is the fibrebasis (or the frame) for the fibre Yx of Y, which is associated with the bundle atlas

.

By virtue of Theorem 1.1.2, vector bundles have global sections, e.g., the global

zero section 0(X). If there is no risk of confusion, we write 0, instead of 0(X).A morphism of vector bundles : Y Y is defined as a fibred morphism over

f : X X whose restriction x : Yx Yf(x) to each fibre of Y is a linear map. Itis called a linear bundle morphism over f.

The following assertion is a corollary of Theorem 1.1.3.

Proposition 1.1.4. IfY X and Y X are vector bundles and : Y Y is alinear bundle morphism of constant rank over X, then the image of and the kernel

Ker0 of with respect to the zero section 0 of Y X are vector subbundles ofY X and Y X, respectively. Note that a vector subbundle of a vector bundleis a closed imbedded submanifold. 2

Unless otherwise stated, by Ker of a linear bundle morphism is meant its

kernel with respect to the zero section 0.There are the following standard constructions of new vector bundles from old. Let Y X be a vector bundle with a typical fibre V. By Y X is meant

the dual vector bundle with the typical fibre V, dual of V. The interior

product of Y and Y is defined as a fibred morphism

: Y Y X

X R.


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where (yi) are linear coordinates on the vector bundle Y.By virtue of Theorem 1.1.2, affine bundles have global sections.

A morphism of affine bundles : Y Y is a fibred morphism over f whoserestriction x : Yx Yf(x) to each fibre of Y is an affine map. It is called an affinebundle morphism over f.

Every affine bundle morphism : Y Y from an affine bundle Y modelledover a vector bundle Y to an affine bundle Y modelled over a vector bundle Y

determines uniquely the linear bundle morphism

: Y

Y,

yi = i

yjyj,

called the linear derivative of .

Let Y X

Y be the fibred product of two affine bundles Y X and Y Xwhich are modelled over the vector bundles Y X and Y X, respectively. Thisproduct, called the Whitney sum, is also an affine bundle modelled over Y

XY.

Furthermore, let Y X be an affine bundle modelled over a vector bundle Y X.Let Y

Y be an affine subbundle modelled over a vector bundle Y

X. Assume

that Y is the Whitney sum ofY and a complementary vector bundle Z X. Thenone can easily verify that the affine bundle Y X decomposes in the Whitney sum

Y = Y X

Z.

Tangent and cotangent bundles

The fibres of the tangent bundle

Z : T Z Zof a manifold Z are tangent spaces to Z. Given a coordinate atlas

Z = {(U, )}

of a manifold Z, the tangent bundle is provided with the (holonomic) atlas

= {(1z U), = T )},


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There is the commutative diagram

T YT T X

? ?Y

X

The tangent bundle T Y Y of a fibre bundle Y X has the vertical tangentsubbundle

V Ydef=Ker T

of T Y, given by the coordinate relation x = 0. This subbundle consists of the

vectors tangent to fibres ofY. The vertical tangent bundle V Y is provided with the

coordinates (x, yi, yi) with respect to the frames {i}.Let T be the tangent map to a fibred morphism : Y Y. Its restriction

V = T|V Y : V Y V Y,yi V = Vi = yjji, (1.1.4)

to V Y is a linear bundle morphism of the vertical tangent bundle V Y to the vertical

tangent bundle V Y

, called the vertical tangent map to .Every vector bundle Y X admits the canonical vertical splittingof the vertical

tangent bundle

V Y = Y X

Y (1.1.5)

because the coordinates yi on V Y have the same transformation law as the linear

coordinates yi on Y.

An affine bundle Y X modelled over a vector bundle Y X also admits thecanonical vertical splitting of the vertical tangent bundle

V Y = Y X

Y (1.1.6)

because the coordinates yi on V Y have the same transformation law as the linear

coordinates yi on the vector bundle Y.

The cotangent bundle TY of a fibre bundle Y X is equipped with thecoordinates (x, yi, x, yi). There is its natural fibration T

Y X over X, but notover TX.


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The vertical cotangent bundle VY Y of a fibre bundle Y X is definedas the vector bundle dual of the vertical tangent bundle V Y Y. It should beemphasized that there is no canonical injection of VY into the cotangent bundle

TY of Y, but we have the canonical projection

: TY Y

VY, (1.1.7)

: xdx + yidy

i yidyi,where {dyi} are the bases for fibres of VY, which are dual of the frames {i} inthe vertical tangent bundle V Y.

With V Y and VY, we have the following two exact sequences of vector bundles

over Y:

0 V Y T Y Y X

T X 0, (1.1.8)0 Y

XTX TY VY 0. (1.1.9)

For the sake of simplicity, we will denote the pull-backs

Y X

T X, Y X

TX

simply by T X and TX.

Example 1.1.2. Let us consider the tangent bundle T TX of TX and the cotan-

gent bundle TT X ofT X. Relative to coordinates (x, p = x) on TXand (x, x)

on T X, these fibre bundles are provided with the coordinates (x, p, x, p) and

(x, x, x, x), respectively. By inspection of the coordinate transformation laws,

one can show that there is the isomorphism

: T TX = TT X, p x, p x (1.1.10)of these bundles over T X [43, 96].

Given a fibre bundle Y X, there is the similar isomorphismV : V V

Y = VV Y, pi yi, pi yi (1.1.11)over V Y, where (x, yi, pi, y

i, pi) and (x, yi, yi, yi, yi) are coordinates on V V

Y and

VV Y, respectively.

Sheaves


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There are several equivalent definitions of sheaves [16, 80]. We will start fromthe following. A sheaf on a topological space X is a topological fibre bundle S X whose fibres, called the stalks, are Abelian groups Sx provided with discrete

topology.

A presheafon a topological space X is defined if an Abelian group SU corresponds

to every open subset U X (S = 0) and, for any pair of open subsets V U,there is the homomorphism

rUV : SU SV

such that

rUU = Id SU,

rUW = rVWr

UV, W V U.

Example 1.1.3. Let X be a topological space, SU the additive Abelian group of

all continuous functions on U X, while the homomorphismrUV : SU SV

is the restriction of these functions to V

X. Then

{SU, r

UV

}is a presheaf.

Every presheaf {SU, rUV} on a topological space X yields a sheaf on X whose

stalk Sx at a point x X is the direct limit of the Abelian groups SU, x U, withrespect to the homomorphisms rUV. It means that, for each open neighbourhood U

of a point x, every element s SU determines an element sx Sx, called the germof s at x. Two elements s SU and s SV define the same germ at x if and onlyif there is an open neighbourhood W x such that

rUWs = rVWs

.

For instance, two real functions s and s

on X define the same germ sx if theycoincide on an open neighbourhood of x. The sheaf generated by the presheaf in

Example 1.1.3 is called the sheaf of continuous functions. The sheaf of smooth

functions on a manifold X is defined in a similar way.

Two different presheaves may generate the same sheaf. Conversely, a sheaf de-

fines a presheaf of Abelian groups (U, S) of local sections of the sheaf S. This

presheaf {(U, S), rUV} is called the canonical presheaf of the sheaf S. It is eas-ily seen that the sheaf generated by the canonical presheaf {(U, S), rUV} of the


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sheaf S coincides with S. Therefore, we will further identify sheaves and canonicalpresheaves.

Example 1.1.4. Let Y X be a vector bundle. The germs of its sections makeup the sheaf S(Y) of sections of Y X. The stalk Sx(Y) of this sheaf at a pointx X consists of the germs of sections of Y X in a neighbourhood of x X.The stalk Sx(Y) is a module over the ring C

x (X) of the germs at x X of smooth

functions on X. If we deal with a tangent bundle T X X, the stalk Sx(T X) is aLie algebra with respect to the Lie bracket of vector fields.

1.2 Multivector fields and differential forms

Subsections: Vector fields, 20; Vector fields on fibre bundles, 21; Multivector fields,

23; The SchoutenNijenhuis bracket, 24; Exterior forms, 26; Exterior forms on fibre

bundles, 27; Interior products, 28; Bivector fields and 2-forms, 29; The Lie derivative,

31; Tangent-valued forms, 32; Distributions, 33; Foliations, 35.

Vector fields

A vector field on a manifold Z is defined as a global section of the tangent bundleT Z Z. The set T(Z) of vector fields on Z is both a module over the ring C(Z)of smooth functions on Z and a real Lie algebra with respect to the Lie bracket

[v, u] = (vu uv), u = u, v = v.

A curve c : () Z, () R, in Z is said to be an integral curve of a vector fieldu on Z if

c = u c,

c

(t) = u

(c(t)), t ().Recall that, for every point z Z, there exists a unique integral curve

c : (, ) Z, > 0,of a vector field u through z = c(0).

A vector field u on an imbedded submanifold N Z is said to be a section ofthe tangent bundle T Z Z over N. It should be emphasized that this is not a


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1.2. MULTIVECTOR FIELDS AND DIFFERENTIAL FORMS 21

vector field on a manifold N since u(N) does not belong to T N T X in general.A vector field on a submanifold N Z is called tangent to the submanifold N ifu(N) T N.

Let U Z be an open subset and > 0. By a local 1-parameter group of localdiffeomorphisms of Z defined on (, ) U is meant a mapping

G : (, ) U (t, z) Gt(z) Z

which possesses the following properties:

for each t (, ), the mapping Gt is a diffeomorphism of U onto the opensubset Gt(U) Z;

Gt+t(z) = (Gt Gt)(z) if t + t (, ).

If such a mapping G is defined on R Z, it is called a 1-parameter group of diffeo-morphisms of Z.

Theorem 1.2.1. [100]. Each local 1-parameter group of local diffeomorphisms G

on U

Z defines a local vector field u on U by setting u(z) to be the tangent vector

to the curve s(t) = Gt(z) at t = 0. Conversely, let u be a vector field on a manifold

Z. For each z Z, there exist a number > 0, a neighbourhood U ofz and a uniquelocal 1-parameter group of local diffeomorphisms on (, ) U, which determinesu. 2

In brief, every vector field u on a manifold Z is the generator of a local 1-

parameter group of local diffeomorphisms. In particular, every exterior form on

a manifold Z is invariant under a local 1-parameter group of local diffeomorphisms

Gu with the generator u, i.e.,

g = , g Gu,

if and only if its Lie derivative Lu along u vanishes.

If a vector field u on a manifold Z is induced by a 1-parameter group of diffeo-

morphisms of Z, then u is called a complete vector field.

Vector fields on fibre bundles


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A vector field u on a fibre bundle Y X is said to be projectable if it projectsover a vector field uX on X, i.e., if the following diagram

Yu T Y

? ?

T

X uX

T X

is commutative. A projectable vector field has the coordinate expression

u = u(x) + ui(x, yj)i, uX = u

.

A vector field = on a base X of a fibre bundle Y X can give rise toa vector field on Y, projectable over , by means of some connection on this fibre

bundle (see (1.4.7) below). Nevertheless, a tensor bundle

T = (m T X) ( k TX),

admits the canonical lift

= + [

1x2m1k + . . . 1x1m2k . . .]

x1m1k(1.2.1)

of any vector field on X. In particular, there exist the canonical lift

= + x x

(1.2.2)

of onto the tangent bundle T X, and its canonical lift

= x x

(1.2.3)

onto the cotangent bundle TX. Hereafter, we will use the compact notation

= x

. (1.2.4)

A projectable vector field u = uii on a fibre bundle Y X is said to be verticalif it projects over the zero vector field uX = 0 on X.

Let Y X be a vector bundle. Using the canonical vertical splitting (1.1.5),we obtain the canonical vertical vector field

uY = yii (1.2.5)


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on Y, called the Liouville vector field. For instance, the Liouville vector field on thetangent bundle T X reads

uTX = x. (1.2.6)

Accordingly, any vector field = on a manifold X has the canonical vertical

lift

V = (1.2.7)

onto the tangent bundle T X.

Multivector fields

A multivector field of degree | |= r (or simply an r-vector field) on a manifoldZ is a section

=1

r!1...r 1 r

of the exterior productr T Z Z. Let us denote by Tr(Z) the vector space of

r-vector fields on Z. In particular, T1(Z) is the space of vector fields on Z (denotedby T(Z) for the sake of simplicity), while T0(Z) is the vector space C(Z) of smoothfunctions on Z. All multivector fields on a manifold Z make up the real Z-graded

vector space T(Z) which is also a Z-graded exterior algebra with respect to theexterior product of multivector fields.

Given a manifold Z, the tangent lift onto T Z of an r-vector field on Z isdefined by the relation(r, . . . , 1) = (r, . . . , 1) (1.2.8)where: (i) k = kdx

are arbitrary 1-forms on the manifold Z, (ii) by

k = xkdx + kdxare meant their tangent lifts (1.2.24) onto the tangent bundle T Z of Z, and (iii)

the right-hand side of the equality (1.2.8) is the tangent lift (1.2.22) onto T Z of thefunction (r, . . . , 1) on Z [67]. We then have the coordinate expression

=1

r!1...r 1 r ,

= 1r!

[z1...r 1 r + (1.2.9)

1...rr

i=1

1 i r ].


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In particular, if is a vector field on a manifold Z, its tangent lift (1.2.9) coincideswith the canonical lift (1.2.2). If an r-vector field is simple, i.e.,

= 1 r,its tangent lift (1.2.9) reads

= ri=1

1V i rV,where kV is the vertical lift (1.2.7) onto T Z of the vector field

k.

Example 1.2.1. The tangent lift of a bivector field

w =1

2w

is

w = 12

(zw + w + w ).

SchoutenNijenhuis bracket

The exterior algebra of multivector fields on a manifold Z is provided with the

SchoutenNijenhuis bracket which generalizes the Lie bracket of vector fields as

follows [13, 181]:

[., .]SN : Tr(M) Ts(M) Tr+s1(M), (1.2.10) =

1

r!1...r 1 r , =

1

s!1...s 1 s ,

[, ]SNdef= + (1)rs ,

=

r

r!s! (2...r

1...s

2 r 1 s ).There following relations hold:

[, ]SN = (1)||||[, ]SN, (1.2.11)[, ]SN = [, ]SN + (1)(||1)|| [, ]SN, (1.2.12)(1)||(||1)[, [, ]SN]SN + (1)||(||1)[, [, ]SN]SN + (1.2.13)

(1)||(||1)[, [, ]SN]SN = 0.


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Example 1.2.2. Let

w =1

2w

be a bivector field. Its SchoutenNijenhuis bracket reads

[w, w]SN = w1w

231 2 3.

The SchoutenNijenhuis bracket commutes with the tangent lift (1.2.9) of mul-tivectors [67], i.e.,

[, ]SN = [, ]SN. (1.2.14)Remark 1.2.3. Let us point out another sign convention used in the definition of

the SchoutenNijenhuis bracket [125]. This bracket, denoted by [., .]SN, is

[, ]SN = (1)||[, ]SN. (1.2.15)

The relation (1.2.11) for this bracket reads

[, ]SN = (1)(||1)(||1)[, ]SN. (1.2.16)

The relation (1.2.12) keeps its form, i.e.,

[, ]SN = [, ]SN + (1)(||1)|| [, ]SN, (1.2.17)

while the relation (1.2.13) is replaced by

(1)(||1)(||1)[, [, ]SN]SN + (1)(||1)(||1)[, [, ]SN]SN + (1.2.18)(1)(||1)(||1)[, [, ]SN ]SN = 0.

The equalities (1.2.16) and (1.2.18) show that, with the modified SchoutenNijenhuis

bracket (1.2.15), the Z-graded vector space T(Z) of multivector fields on a manifoldZ is a graded Lie algebra, where the Lie degree of a multivector field is | | 1.In particular,

ad()def=[, ]SN (1.2.19)


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is a graded endomorphism of degree | | 1 of the graded Lie algebra T(Z). Ifis a vector field, the endomorphism (1.2.19) is the Lie derivative

ad() = L (1.2.20)

of the multivector field along .

Exterior forms

An exterior r-form on a manifold Z is a section

= 1r!

1...r dz1 dzr

of the exterior productr TZ Z. We denote byOr(Z) the vector space of exterior

r-forms on a manifold Z. This is also a module over the ring O0(Z) = C(Z).

From now on we will use the notation C(Z) for the ring of smooth functions on

a manifold Z, while O0(Z) stands for the vector space of these functions as a rule.

All exterior forms on Z constitute the exterior Z-graded algebra O(Z) with

respect to the exterior product. The exterior differentialis the first order differential

operator

d : Or(Z) Or+1(Z),d =

1

r!1...r dz

dz1 dzr ,

on O(Z). It obeys the relations

d d = 0,d( ) = d() + (1)|| d(),

where | | is the degree of .Given a manifold map f : Z Z

, by f

is meant the pull-back on Z of anr-form on Z by f, which is defined by the condition

f(v1, . . . , vr)(z) = (T f(v1), . . . , T f (vr))(f(z)), v1, vr TzZ.We have the relations

f( ) = f f,df = f(d).


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For instance, if iN : N Z is a submanifold, the pull-back iN onto N is calledthe restriction of an exterior form to N.

Exterior forms on fibre bundles

Let : Y X be a fibre bundle with fibred coordinates (x, yi). The pull-backon Y of exterior forms on X by provides the inclusion

: O(X) O(Y).Exterior forms

: Y r TX, =

1

r!1...r dx

1 dxr ,

on Y such that = 0 for arbitrary vertical vector field on Y are said to behorizontal forms. A horizontal n-form is called a horizontal density. We will use the

notation

= dx1 dxn, = . (1.2.21)

In the case of the tangent bundle T X X, there is a different way, besidesthe pull-back, to lift onto T X the exterior forms on X [67, 110, 189]. Let f be a

function on X. Its tangent lift onto T X is defined as the function

f = xf. (1.2.22)Let be an r-form on X. Its tangent lift onto T X is said to be the r-form givenby the relation

(

1, . . . ,

r) =

(1, . . . , r), (1.2.23)where i are arbitrary vector fields on T X, and i are their canonical lifts (1.2.2)onto T X. We have the coordinate expression

=1

r!1r dx

1 dxr ,

= 1r!

[x1r dx1 dxr + (1.2.24)

ri=1

1r dx1 dxi dxr ].


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The following equality holds:

d = d.Example 1.2.4. Given a 2-form

=1

2dx

dx

on a manifold X, its tangent lift (1.2.24) onto T X reads

= 12

(xdx dx + dx dx + dx dx). (1.2.25)

Interior products

The interior product (or the contraction) of a vector field u = u and an

exterior r-form is given by the coordinate expression

u =r

k=1

(

1)k1

r! uk 1...k...r dz1 dzk dzr = (1.2.26)

1

(r 1)! u2...r dz

2 dzr .

It satisfies the relations

(u1, . . . , ur) = ur u1, (1.2.27)u( ) = u + (1)|| u, (1.2.28)[u, u] = ud(u) ud(u) uud, O1(Z). (1.2.29)

The generalization of the interior product (1.2.26) for multivector fields is the

left interior product

= (), | || |, O(Z), T(Z),

of multivector fields and exterior forms, which is derived from the equality

(u1 ur) = (u1, . . . , ur), O(Z), ui T(Z),


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for simple multivector fields. We have the relation

= ( ) = (1)||||, O(Z), , T(Z).

Example 1.2.5. The formula (1.2.29) can be generalized for multivector fields as

follows [13]:

[, ]SN = (1)||(||1)d() + (1)||d() d,where | |=| | + | | 1.

The right interior product = (), | || |, O(Z), T(Z),

of exterior forms and multivector fields is given by the equalities

(1, . . . , r) = r 1, i O1(Z), Tr(Z), = 1

(r 1)! 1...r11 r1, O1(Z).

It satisfies the relations

(

)

=

(

) + (

1)||(

)

, O1(Z),

( ) = , , O(Z).In particular, if | |=| |, we have the natural pairing

, : Tr(Z) Or(Z) C(Z),, = = = () = (). (1.2.30)

Bivector fields and 2-forms

Each bivector field

w =1

2w

on a manifold Z defines the linear fibred morphism

w : TZ Z

T Z,

w()def= w(z), Tz Z, (1.2.31)

w() = w(z),


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which fulfills the relation

w(z)(, ) = w() = w(z), z Z, , Tz Z.One says that a bivector field w is of rank r at a point z Z if the morphism(1.2.31) has rank r at z. If this morphism is an isomorphism at all the points z Z,the bivector field w is said to be non-degenerate. Such a bivector field can exist only

on an even-dimensional manifold.

The morphism (1.2.31) can be generalized to the homomorphism of graded al-

gebras O(Z) T(Z) in accordance with the relationw()(1, . . . , r)

def=(1)r(w(1), . . . , w(r)), (1.2.32)

Or(Z), i O1(Z).This is clearly an isomorphism if the bivector field w is non-degenerate.

Each 2-form

=1

2dz

dz

on a manifold Z defines the linear fibred morphism

: T Z TZ,(v)

def= v(z), v TzZ, (1.2.33)

(v) = (z)vdz.One says that a 2-form is of rank r at a point z Z if the morphism (1.2.33) hasrank r at z. This is the maximal number 2k such that

k (z) = 0.The kernel of a 2-form is defined as the kernel

Kerdef=zZ

{v TzZ : vu = 0, u TzZ} (1.2.34)

of the morphism (1.2.33). Its fibre Ker z at a point z Z is a vector subspace ofthe tangent space TzZ whose codimension equals the rank of at z. If a 2-form

is of constant rank, its kernel (1.2.34) is a subbundle of the tangent bundle T Z in

accordance with Proposition 1.1.4.

A 2-form is called non-degenerate if its rank is equal to dim Z at all points

z Z. A non-degenerate 2-form can exist only on a 2m-dimensional manifold.Then

m is nowhere vanishing, and can play the role of a volume element on Z.


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On a 2m-dimensional manifold Z, there is one-to-one correspondence betweenthe non-degenerate 2-forms w and the non-degenerate bivector fields w in accor-

dance with the equalities

w(, ) = w(w(), w

()), (1.2.35)

w(, ) = w(w(),

w()), (1.2.36)

, O1(Z), , T(Z),where the morphisms w (1.2.31) and

w (1.2.33) obey the relations

w = (w)1,ww

=

,

i.e.,

w(w()) = ,

w(w

()) = .

The Lie derivative

The Lie derivative of an exterior form along a vector field u is given by the

equality

Lu = ud + d(u).In particular, if f is a function, then

Luf = u(f) = udf.The relation

Lu(

) = Lu

+

Lu

is fulfilled. Given the tangent lift (1.2.24) of an exterior form , we haveLu() = u

[67, 147]. The Lie derivative (1.2.20) of a multivector field along a vector field u

is

Lu = [u, ]SN = [u, ]SN,


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and it obeys the equality

Lu( ) = Lu + Lu

in accordance with the relation (1.2.17).

Tangent-valued forms

Elements of the tensor product Or(Z) T(Z) are called the tangent-valuedr-forms

: Z r TZ T Z, =

1

r!1...r dz

1 dzr .

There is one-to-one correspondence between the tangent-valued 1-forms on a ma-

nifold Z and the linear bundle endomorphisms over Z:

: T Z T Z,

: TzZ v v(z) TzZ, (1.2.37)

and : TZ TZ, : Tz Z v (z)v Tz Z. (1.2.38)In particular, the canonical tangent-valued 1-form

Z = dz

on Z corresponds to the identity morphisms (1.2.37) and (1.2.38).

Let Z = T X. There is the fibred endomorphism J of the tangent bundle T T X

of T X such that, for every vector field on X, we have

J = V, J V = 0,where is the canonical lift (1.2.2) and V is the vertical lift (1.2.7) onto T T X of avector field on T X. This endomorphism reads

J() = , J() = 0. (1.2.39)


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It is readily observed that JJ = 0, and the rank ofJ equals n. The endomorphismJ (1.2.39), called an almost tangent structure[110, 189], corresponds to the tangent-

valued form

J = dx (1.2.40)

on the tangent bundle T X.

Distributions

An n-dimensional smooth distribution on a k-dimensional manifold Z is an n-dimensional subbundle T of the tangent bundle T Z. We will say that a vector field

v on Z is subordinateto a distribution T if it is a section of T Z. An integralcurve of a vector field, subordinate to a distribution T, is called admissible with

respect to T.

A distribution T is said to be involutive if the Lie bracket [u, u] is a section of

T Z, whenever u and u are sections of the distribution T Z.A connected submanifold N of a manifold Z is called an integral manifold of a

distribution T on Z if the tangent spaces to N belong to the fibres of this distribution

at each point of N. Unless otherwise stated, by an integral manifold we mean an

integral manifold of maximal dimension, equal to dimension of the distribution T.

An integral manifold N is called maximalif there is no other integral manifold which

contains N.

Theorem 1.2.2. [185]. Let T be a smooth involutive distribution on a manifold Z.

For any point z Z, there exists a unique maximal integral manifold of T passingthrough z. 2

In view of this fact, involutive distributions are also called completely integrable

distributions.If a distribution T is not involutive, there are no integral submanifolds of di-

mension equal to the dimension of a distribution. However, integral submanifolds

always exist, e.g., the integral curves of vector fields, subordinate to T.

We refer the reader to [68] for a detailed exposition of differential and Pfaffian

systems.

A differential system S on a manifold Z is said to be a subbundle of the sheaf

S(T Z) of vector fields on Z whose fibre Sz at each point z Z is a submodule


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of the Cz (Z)-module Sz(T Z) (see Example (1.1.4)). The germs of sections of adistribution T obviously make up a differential system S(T).

The flag of a differential system S is the sequence of differential systems

S1 = S, S2 = [S, S], Si = [Si1, S].Here [S, S]z is the C

z (Z)-module generated by [v, u], v Sz, u Sz. Let S(T) be

a differential system associated with a distribution T, and let

S(T) = S1 S2

be its flag. In general, Si is not associated with a distribution. If this is the case forall i, we may define the flag of a distribution

T = T1 T2 . (1.2.41)A distribution is called regular if its flag (1.2.41) is well defined. The sequence

(1.2.41) stabilizes, i.e., there exists an integer r such that Tr1 = Tr = Tr+1 [184];moreover Tr is involutive. In particular, if r = 1, we are dealing with the integrable

case. If Tr = T Z, the distribution T is called totally non-holonomic.

A codistribution T on a manifold Z is a subbundle of the cotangent bundle.

For instance, the annihilator Ann T of an n-dimensional distribution T is a (k n)-dimensional codistribution.

Theorem 1.2.3. [185]. Let T be a distribution and Ann T its annihilator. Let

Ann T be the ideal of the exterior algebra O(Z) which is generated by elements ofAnn T. A distribution T is involutive if and only if the ideal Ann T is a differentialideal, i.e., d(Ann T) Ann T. 2

Corollary 1.2.4. Let T be a smooth involutive r-dimensional distribution on a

k-dimensional manifold Z. Every point z Z has an open neighbourhood U zwhich is a domain of a coordinate chart (z1, . . . , zk) such that the restrictions of the

distribution T and its annihilator Ann T to U are generated by the r vector fields

z1, . . . ,

z r

and the (k r) 1-forms dzkr+1, . . . , d zk, respectively. It follows that integral man-ifolds of an involutive distribution make up a foliation. 2


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Example 1.2.6. Every 1-dimensional distribution on a manifold Z is integrable.Its section is a nowhere vanishing vector field u on Z, while its integral manifolds are

the integral curves of u. By virtue of Corollary 1.2.4, there exist local coordinates

(z1, . . . , zk) around each point z Z such that u is given by

u =

z1.

A Pfaffian system S

is a submodule of the C

(Z)-module O

1

(Z). In particular,sections of a codistribution constitute a Pfaffian system. Any Pfaffian system S

defines the ideal S of the exterior algebra O(Z) which is generated by elementsof S.

Given a flag (1.2.41) of a regular distribution, one can introduce the coflag of

the codistribution

Ann(T) Ann(T2) . (1.2.42)

The coflag (1.2.42) stabilizes. In particular, a distribution T is totally non-holonomic

if and only if its coflag (1.2.42) shrinks to zero.

Foliations

An r-dimensional (regular) foliation on a k-dimensional manifold Z is said to be

a partition of Z into connected leaves F with the following property. Every point

of Z has an open neighbourhood U which is a domain of a coordinate chart (z)

such that, for every leaf F, the connected components F U are described by theequations

zr+1 = const., zk = const.

[90, 150]. Note that leaves of a foliation fail to be imbedded submanifolds, i.e.,

topological subspaces in general.

Example 1.2.7. Submersions : Y X and, in particular, fibre bundles arefoliations with the leaves 1(x), x (Y) X. A foliation is called simple if it isa fibre bundle. Any foliation is locally simple.


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Example 1.2.8. Every real function f on a manifold Z with nowhere vanishingdifferential df is a submersion Z R. It defines a 1-codimensional foliation whoseleaves are given by the equations

f(z) = c, c f(Z) R.

This is the foliation of level surfacesof the function f, called a generating function.

Every 1-codimensional foliation is locally a foliation of level surfaces of some function

on Z.

The level surfaces of arbitrary function f = const. on a manifold Z define a sin-gular foliation F on Z [90]. Its leaves are not submanifolds in general. Nevertheless

if df(z) = 0, the restriction of F to some open neighbourhood U of z is a foliationwith the generating function f|U.

1.3 Jet manifolds

Subsections: Jet manifolds, 36; Canonical horizontal splittings, 38; Second order jet

manifolds, 39; The total derivative, 41; Higher order jet manifolds, 41; Differential

operators and differential equations, 41.

Jet manifolds

Given a fibre bundle Y X with bundle coordinates (x, yi), let us considerthe equivalence classes j1xs, x X, of its sections s, which are identified by theirvalues si(x) and the values of their first derivatives s

i(x) at points x X. Theequivalence class j1xs is called the first order jet of sections s at the point x X.The set J1Y of first order jets is provided with a manifold structure with respect to

the adapted coordinate atlas

(x, yi, yi),

(x, yi, yi)(j1xs) = (x

, si(x), si(x)),

yi =

x

x( + y

jj)y

i. (1.3.1)

It is called the jet manifold of sections of the fibre bundle Y X (or simply the jetmanifold of the fibre bundle Y X).


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1.3. JET MANIFOLDS 37

The jet manifold admits the natural fibrations

1 : J1Y j1xs x X, (1.3.2)10 : J

1Y j1xs s(x) Y, (1.3.3)where (1.3.3) is an affine bundle modelled over the vector bundle

TXY

V Y Y.

For the sake of convenience, the fibration J1Y X is further called a jet bundle,while the fibration J1Y

Y is an affine jet bundle.

There are the following two canonical monomorphisms of the jet manifold J1Y

over Y:

: J1Y TXY

T Y, (1.3.4)

= dx d = dx ( + yii),where d is called the total derivative, and

1 : J1Y TY

YV Y, (1.3.5)

1 = i

i = (dyi

yi

dx

) i,where

i = dyi yidx (1.3.6)is called the contact form. In accordance with these monomorphisms, every element

of the jet manifold J1Y can be represented by the tangent-valued forms

dx ( + yii) and (dyi yidx) i.Each fibred morphism : Y

Y over a diffeomorphism f is extended to the

fibred morphism of the corresponding jet manifolds

J1 : J1Y

J1Y,

J1 : j1xs j1f(x)( s f1),

yi J1 = (jiyj + i)

(f1)

x ,

called the jet prolongation of the morphism .


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Each section s of a fibre bundle Y X has the jet prolongation to the section(J1s)(x)

def= j1xs,

(yi, yi) J1s = (si(x), si(x)),of the jet bundle J1Y X. A section s of the jet bundle J1Y X is said to beholonomic if this is the jet prolongation of some section of the fibre bundle Y X.

Any projectable vector field

u = u(x) + ui(x, yj)i

on a fibre bundle Y X admits the jet prolongation to the vector fieldu = r1 J1u : J1Y J1T Y T J1Y,u = u + u

ii + (dui yiu)i , (1.3.7)

on the jet manifold J1Y. One can show that the jet prolongation of vector fields

u u is the morphism of Lie algebras, i.e.,[u, u] = [u, u].

In order to obtain (1.3.7), we have used the canonical fibred morphism

r1 : J1T Y T J1Y,

yi r1 = (yi) yix.In particular, there is the canonical isomorphism

V J1Y = J1V Y, (1.3.8)

yi = (yi).

Canonical horizontal splittings

The canonical morphisms (1.3.4) and (1.3.5) can be viewed as the morphisms

: J1Y X

T X d = J1Y Y

T Y (1.3.9)

and

1 : J1Y Y

VY dyi i = 1dyi J1Y Y

TY, (1.3.10)


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where {dyi} are the bases for the fibres of the vertical cotangent bundle VY. Thesemorphisms determine the canonical horizontal splittingsof the pull-backs

J1Y Y

T Y = (T X) J1Y

V Y, (1.3.11)

x + yii = x

( + yii) + (y

i xyi)i,and

J1Y Y

TY = TX J1Y

1(VY), (1.3.12)xdx

+ yidyi = (x + yiy

i)dx

+ yi(dyi

yidx

).

Second order jet manifolds

Taking the first order jet manifold of the jet bundle J1Y X, we come to therepeated jet manifold J1J1Y, provided with the adapted coordinates

(x, yi, yi, yi(), y

i),

yi() =

x

x ( + y

j()j)y

i,

yi =

x

x

( + yj()j + y

j

j )y

i.

There exist two different affine fibrations of J1J1Y over J1Y:

the familiar affine jet bundle (1.3.3)

11 : J1J1Y J1Y, yi 11 = yi, (1.3.13)

modelled over the vector bundle

TX J1Y

V J1Y J1Y, (1.3.14)

and the affine bundleJ110 : J

1J1Y J1Y, yi J110 = yi(), (1.3.15)

whose underlying vector bundle

J1(TX V Y) J1Y (1.3.16)

differs from (1.3.14).


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In general, there is no canonical identification of these fibrations, but it can be madeby means of a symmetric linear connection on X [57].

The points q J1J1Y, where 11(q) = J110(q), make up the affine subbundleJ2Y J1Y of J1J1Y, called the sesquiholonomic jet manifold. This is given bythe coordinate conditions

yi() = yi,

and is coordinated by (x, yi, yi, yi).

The second order jet manifold J2Y of a fibre bundle Y X is the affine sub-bundle 21 : J2Y J1Y of the fibre bundle J2Y J1Y, given by the coordinateconditions

yi = yi

and coordinated by (x, yi, yi, yi = y

i). It is modelled over the vector bundle

2 TX J1Y

V Y J1Y.

The second order jet manifold J2Y can also be seen as the set of the equivalence

classes j2

xs of sections s of the fibre bundle Y X, which are identified by theirvalues and the values of their first and second order partial derivatives at pointsx X:

yi(j2xs) = s

i(x), yi(j2xs) = s

i(x).

Let s be a section of a fibre bundle Y X and J1s its jet prolongation to asection of the jet bundle J1Y X. The latter gives rise to the section J1J1s ofthe repeated jet bundle J1J1Y X. This section takes its values into the secondorder jet manifold J2Y. It is called the second order jet prolongation of the section

s, and is denoted by J

2

s.

Proposition 1.3.1. Let s be a section of the jet bundle J1Y X and J1s its jetprolongation to the section of the repeated jet bundle J1J1Y X. The followingthree facts are equivalent:

s = J1s where s is a section of the fibre bundle Y X; J1s takes its values into J2Y;


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J1s takes its values into J2Y.2

The total derivative

We will use the total derivative operator

d = + yii + y

i

i .

It satisfies the equalities

d( ) = d() + d(),d(d) = d(d()).

Higher order jet manifolds

The k-order jet manifold JkY of a fibre bundle Y X comprises the equivalenceclasses jkxs, x X, of sections s of Y identified by the k + 1 terms of their Tailorseries at the points x

X. The jet manifold JkY is provided with the adapted

coordinates

(x, yi, yi, . . . , yik1

),

yil1(jkxs) = l 1si(x), 0 l k.

Every section s of a fibre bundle Y X gives rise to the section Jks of the fibrebundle JkY X such that

yil1 Jks = l 1si, 0 l k.

Differential operators and differential equations

Let JkY be the k-order jet manifold of a fibre bundle Y X and E X avector bundle over X.

Definition 1.3.2. A fibred morphism

E: JkY X

E (1.3.17)


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is called a k-order differential operator on the fibre bundle Y X. It sends eachsection s(x) of Y X onto the section (E Jks)(x) of the vector bundle E X.2

The kernel of a differential operator is the subset

Ker E= E1(0(X)) JkY, (1.3.18)where 0 is the zero section of the vector bundle E X, and we assume that

0(X) E(JkY).

Definition 1.3.3. A system of k-order partial differential equations (or simply adifferential equation) on a fibre bundle Y X is defined as a closed subbundle Eof the jet bundle JkY X [20, 57, 104]. 2

Its (classical) solution is a (local) section s of the fibre bundle Y X such thatits k-order jet prolongation Jks lives in E.

For instance, if the kernel (1.3.18) of a differential operator E is a closed sub-bundle of the fibre bundle JkY X, it defines a differential equation

E Jks = 0.

The following condition is sufficient for a kernel of a differential operator to bea differential equation.

Proposition 1.3.4. Let the morphism (1.3.17) be of constant rank. By virtue of

Theorem 1.1.3, its kernel (1.3.18) is a closed subbundle of the fibre bundle JkY Xand, consequently, is a k-order differential equation. 2

1.4 Connections

Subsections: Connections, 42; The curvature of connections, 44; Linear connections,44; Affine connections, 45; Flat connections, 45.

Connections

A connection on a fibre bundle Y X is defined as a global section : Y J1Y, = dx ( + i(x, yj)i),


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1.4. CONNECTIONS 43

of the affine jet bundle J1Y Y. Combining a connection and the morphisms(1.3.9) and (1.3.10) gives the splittings

: T X T Y,1 : VY TYof the exact sequences (1.1.8) and (1.1.9), respectively. Accordingly, substitution of

the section yi = i into the expressions (1.3.11) and (1.3.12) leads to the familiar

splittings of the tangent bundle

T Y = (T X) Y V Y, (1.4.1)x + y

ii = x( +

ii) + (y

i xi)i,and the cotangent bundle

TY = TXY

(VY), (1.4.2)

xdx + yidy

i = (x + iyi)dx

+ yi(dyi idx),

of a fibre bundle Y X with respect to the connection . In an equivalent way,the connection defines the corresponding projection

: T Y x + yii (yi xi)i V Y (1.4.3)and the corresponding section

= (dyi idx) i (1.4.4)of the fibre bundle TY

YV Y Y.

Connections on a fibre bundle Y X constitute an affine space modelled overthe linear space ofsoldering forms

: Y

TXY

V Y,

= idx i.

Any connection on a fibre bundle Y X defines the first order differentialoperator on Y

D : J1Y z [z (10(z))] TX

YV Y, (1.4.5)

D = (yi i)dx i,


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called the covariant differential. Its action on sections s of the fibre bundle Y reads

s = D J1s = [si ( s)i]dx i. (1.4.6)For instance, a section s is said to be an integral section for a connection , if

s = 0, i.e., s = J1s. For any section s of a fibre bundle Y X, there existsa connection on Y X such that s is its integral section. This connection is anextension of the section s(x) J1s(x) of the affine jet bundle J1Y Y over theclosed submanifold s(X) Y in accordance with Theorem 1.1.2.

A connection on a fibre bundle Y X defines the horizontal lift = ( + ii) (1.4.7)

onto Y of each vector field = on X.

The curvature of connections

The curvature of a connection on a fibre bundle Y X is said to be the2-form on Y

R : Y 2 TXY

V Y,

R =1

2Ridx dx i,

Ri = i i + jji jji. (1.4.8)

Linear connections

Let Y X be a vector bundle. A linear connection on Y X reads = dx [ + ij(x)yji].

It defines the dual linear connection

= dx [ ij(x)yji]on the dual vector bundle Y X. For instance, a linear connection K on thetangent bundle T X, and the dual linear connection K on the cotangent bundle

TX are given by the expressions

K = dx ( + K(x)x), (1.4.9)K = dx ( K(x)x). (1.4.10)


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1.4. CONNECTIONS 45

Affine connections

Let Y X be an affine bundle modelled over a vector bundle Y X. Anaffine connection on Y X reads

= dx [ + (ij(x)yj + i(x))i].It defines the linear connection

= dx [ + ij(x)yj y i

]

on the vector bundle Y X.Flat connections

Each connection on a fibre bundle Y X, by definition, yields the horizontaldistribution (T X) T Y on Y, generated by the horizontal vector fields (1.4.7).The following assertions are equivalent.

The horizontal distribution is involutive.

The connection is flat (curvature-free), i.e., its curvature is equal to zero

everywhere.

There is an integral section for the connection through any point y Y.Hence, a flat connection on Y X yields the integrable horizontal distribution,i.e., the horizontal foliation on Y, transversal to the fibration Y X. Its leafthrough a point y Y is defined locally by an integral section sy for the connection through y. Conversely, let a fibre bundle Y X admit a horizontal foliation suchthat, for each point y Y, the leaf of this foliation through y is locally defined bya section sy of Y

X through y. Then the map

: Y J1Y,(y) = j1xsy, (y) = x,

introduces a flat connection on Y X. Thus, there is one-to-one correspondencebetween the flat connections and the horizontal foliations on a fibre bundle Y X.

Given a horizontal foliation on a fibre bundle Y X, there exists the associatedatlas of bundle coordinates (x, yi) ofY such that every leaf of this foliation is locally


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generated by the equations yi = const., and the transition functions yi yi(yj) areindependent of the base coordinates x [23, 57]. This is called the atlas of constant

local trivializations. Two such atlases are said to be equivalent if their union is

also an atlas of constant local trivializations. They are associated with the same

horizontal foliation. Thus, we come to the following assertion.

Proposition 1.4.1. There is one-to-one correspondence between the flat connec-

tions on a fibre bundle Y X and the equivalence classes of atlases of constantlocal trivializations of Y such that i = 0 relative to these atlases. 2

1.5 Bundles with symmetries

Subsections: Tangent and cotangent bundles of Lie groups, 46; Principal bundles,

48; The linear frame bundle, 52.

Tangent and cotangent bundles of Lie groups

Let G be a real Lie group with dim G > 0 and gl [gr] its left [right] Lie algebra

of left-invariant vector fields l(g) = T Lg(l(e)) [right-invariant vector fields r(g) =T Rg(r(e))] on the group G. Here, e is !

the unit element of G, while Lg and Rg denote the action of G on itself on

the left and on the right, respectively. Every left-invariant vector field l(g) [right-

invariant vector field r(g)] corresponds to the element v = l(e) [v = r(e)] of the

tangent space TeG provided with both left and right Lie algebra structures. For

instance, given v TeG, let vl(g) and vr(g) be the corresponding left-invariant andright-invariant vector fields. There is the relation

vl(g) = T Lg T R1g (vr(g)).

Let {m = m(e)} [{m = m(e)}] denote the basis for the left [right] Lie algebra,and let ckmn be the right structure constants:

[m, n] = ckmnk.

The mapping g g1 yields the isomorphism

: gl m m = m gr (1.5.1)


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1.5. BUNDLES WITH SYMMETRIES 47

of left and right Lie algebras. For instance, we have

[m, n] = ckmnk.The tangent bundle G : T G G of the Lie group G is trivial. There are the

isomorphisms

l : T G q (g = G(q), T L1g (q)) G gl,r : T G q (g = G(q), T R1g (q)) G gr.

The left action Lg of a Lie group G on itself defines its adjoint representation g

Adg in the right Lie algebra gr and its identity representation in the left Lie algebragl. Correspondingly, there is the adjoint representation

: ad () = [, ],ad m(n) = c

kmnk,

of the right Lie algebra gr in itself.

An action

G Z (g, z) gz Zof a Lie group G on a manifold Z on the left yields the homomorphism

gr T(Z)of the right Lie algebra gr of G into the Lie algebra of vector fields on Z such that

Ad g() = T g g1 (1.5.2)[100]. Vector fields m are said to be the generators of a representation of the Lie

group G in Z.

Let g

= T

e G be the vector space dual of the tangent space TeG. It is called thedual Lie algebra (or the Lie coalgebra), and is provided with the basis {m} dual ofthe basis {m} for TeG. The group G and the right Lie algebra gr act on g by thecoadjoint representation

Adg(), def=, Ad g1(), g, gr, (1.5.3)ad(), = , [, ], gr,adm(

n) = cnmkk.


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Remark 1.5.1. In the literature (see, e.g.,[2]), one can meet another definition ofthe coadjoint representation in accordance with the relation

Adg(), = , Ad g().

An exterior form on the group G is said to be left-invariant [right-invariant] if

(e) = Lg((g)) [(e) = Rg((g))]. The exterior differential of a left-invariant [right-

invariant] form is left-invariant [right-invariant]. In particular, the left-invariant

1-forms satisfy the MaurerCartan equations

d(, ) = 12

([, ]), , gl.

There is the canonicalgl-valued left-invariant 1-form

l : TeG glon a Lie group G. The components ml of its decomposition l =

ml m with respect

to the basis for the left Lie algebra gl make up the basis for the space of left-invariant

exterior 1-forms on G:

mnl = nm.The MaurerCartan equation, written with respect to this basis, reads

dml =1

2cmnk

nl kl .

Accordingly, the canonicalgr-valued right-invariant 1-form

r : TeG gron the group G is defined. There are the relations

l(vg) = l(T L1g (vg)) = T L

1g (vg), vg TgG,

r(vg) = r(T R1g (vg)) = T R

1g (vg),

(l(vg)) = T Lg T R1g r(vg) = Adg(r(vg)),where is the isomorphism (1.5.1).

Principal bundles


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We refer the reader to [100, 170, 192] for the general theory of principal bundles.Let P : P X be a principal bundlewith a real structure Lie group G. There

is the canonical free transitive action

RG : PX

G P, (1.5.4)Rg : p pg, p P, g G,

of G on P on the right.

A principal bundle P is equipped with a bundle atlas P = {(U, P )} whosetrivialization morphisms

P : 1P (U) U G

obey the condition

pr2 P Rg = g pr2 P , g G.

Due to this property, every trivialization morphism P uniquely determines a local

section z : U P such that

pr2 P

z = e.The transformation rules for z read

z(x) = z(x)(x), x U U, (1.5.5)

where are the transition functions of the atlas P. Conversely, the family

{(U, z)} of local sections of P, which obey (1.5.5), uniquely determines a bundleatlas P of P.

A principal bundle P X admits the canonical trivial vertical splitting

: V P = P glsuch that 1(m) are fundamental vector fields on P corresponding to the basis

elements m for the left Lie algebra gl.

Taking the quotient of the tangent bundle T P P and the vertical tangentbundle V P of P by the tangent map T Rg, we obtain the vector bundles

TGP = TP/G and VGP = V P/G (1.5.6)


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over X. Sections of TGP X are G-invariant vector fields on P, while sections ofVGP X are G-invariant vertical vector fields on P. Hence, the typical fibre ofVGP X is the right Lie algebra gr of the right-invariant vector fields on the groupG. The group G acts on this typical fibre by the adjoint representation.

The Lie bracket of vector fields on P goes to the quotients (1.5.6) and defines the

Lie bracket of sections of the vector bundles TGP X and VGP X. It followsthat VGP X is a fibre bundle of Lie algebras (the gauge algebra bundle in theterminology of gauge theories) whose fibres are isomorphic to the right Lie algebra

gr of the group G.

Example 1.5.2. When P = X G is trivial, we haveVGP = X TG/G = X gr.

Example 1.5.3. Given a local bundle splitting of P, there are the corresponding

local bundle splitting of TGP and VGP. Given the basis {p} for the Lie algebra gr,we obtain the local fibre bases {, p} for the fibre bundle TGP X and {p} forthe fibre bundle VGP X. If

, : X TGP, = +

pp, = +

qq,

are sections, the coordinate expression of their bracket is

[, ] = ( ) + (r r + crpqpq)r.

Let J1P be the first order jet manifold of a principal bundle P X with a

structure Lie group G. Bearing in mind that the jet bundle J1

P P is an affinebundle modelled over the vector bundle

TXP

V P P,

let us consider the quotient of the jet bundle J1P P by the jet prolongation J1Rgof the canonical action (1.5.4). We obtain the affine bundle

C = J1P/G X (1.5.7)


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modelled over the vector bundle

C = TX VGP X.Hence, there is the canonical vertical splitting

V C = CX

C.

In the case of a principal bundle P X, the exact sequence (1.1.8) reduces tothe exact sequence

0

VGP

XTGP

T X

0. (1.5.8)

A principal connection A on a principal bundle P X is defined as a sectionA : P J1P which is equivariant under the action (1.5.4) of the group G on P,i.e.,

J1Rg A = A Rg, g G. (1.5.9)Turning now to the quotients (1.5.6), such a connection defines the splitting of the

exact sequence (1.5.8). It is represented by the tangent-valued form

A = dx ( + Aqq), (1.5.10)where Ap are local functions on X.

On the other hand, due to the property (1.5.9), there is obviously one-to-one

correspondence between the principal connection on a principal bundle P X andthe global sections of the fibre bundle C X (1.5.7), called the bundle of principalconnections.

Let a principal connection on the principal bundle P X be represented bythe vertical-valued form A (1.4.4). Then the form

A : PA TP

PV P

Id TP glis the familiar gl-valued connection form on the principal bundle P. Given a localbundle splitting (U, z) of P, this form reads

A = P Aqdx q,where P is the canonical gl-valued 1-form on P, {p} is the basis ofgl, and Ap arelocal functions on P such that

Aq(pg)q = A

q(p)Adg

1(q).


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The pull-back zA of A over U is the well-known local connection 1-form

A = Aqdx q, (1.5.11)where Aq = A

q z are local functions on X.

It is readily observed that the coefficients Aq of this form are precisely the co-

efficients of the form (1.5.10). Moreover, given a bundle atlas of P, the bundle of

principal connections C is equipped with the associated bundle coordinates (x, aq)

such that, for any section A of C X, the local functionsAq = a

q

A

are again the coefficients of the local connection 1-form (1.5.11). In gauge theory,

these coefficients are treated as gauge potentials. We will use this term to refer to

sections A of the fibre bundle C X.Let now

Y = (P V)/G (1.5.12)be a fibre bundle associated with the principal bundle P X whose structuregroup G acts on the typical fibre V of Y on the left. Let us recall that the quotient

in (1.5.12) is defined by identification of the elements (p, v) and (pg,g1v) for all

g G. Briefly, we will say that (1.5.12) is a P-associated fibre bundle.As is well known, the principal connection A (1.5.10) induces the corresponding

connection on the P-associated fibre bundle (1.5.12). If Y is a vector bundle, this

connection takes the form

A = dx ( + ApIipi),where Iip are generators o

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