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Geometry in a Fr´echet Context:

A Projective Limit Approach

by

C.T.J. Dodson

University of Manchester,

Manchester, UK

George Galanis

Hellenic Naval Academy,

Piraeus, Greece

Efstathios Vassiliou

National and Kapodistrian University of Athens,

Athens, Greece

Contents

Preface page vii

1 Banach manifolds and bundles 1

1.1 Banach manifolds 1

1.2 Banach-Lie groups 7

1.3 Smooth actions 12

1.4 Banach vector bundles 14

1.5 Connections on vector bundles 27

1.6 Banach principal bundles 37

1.7 Connections on principal bundles 53

1.8 The curvature of a principal connection 63

1.9 Holonomy groups 67

1.10 Classiﬁcation of ﬂat bundles 71

2 Fr´echet spaces 73

2.1 The topology of Fr´echet spaces 73

2.2 Diﬀerentiability 79

2.3 Fr´echet spaces as pro jective limits 82

2.4 Diﬀerential equations in Fr´echet spaces 99

3 Fr´echet manifolds 105

3.1 Smooth structures on Fr´echet manifolds 106

3.2 The tangent bundle of a plb-manifold 111

3.3 Vector ﬁelds 119

3.4 Fr´echet-Lie groups 121

3.5 Equations with Maurer-Cartan diﬀerential 127

3.6 Diﬀerential forms 131

4 Projective systems of principal bundles 139

4.1 Projective systems and Fr´echet principal bundles 139

4.2 Connections on limit principal bundles 150

v

vi Contents

4.3 Parallel translations and holonomy groups 160

4.4 The curvature of a plb-connection 165

4.5 Flat plb-bundles 172

5 Projective systems of vector bundles 183

5.1 A particular Fr´echet group 183

5.2 Projective systems and Fr´echet vector bundles 185

5.3 Morphisms of plb-vector bundles 192

5.4 The sections of plb-vector bundles 198

5.5 The pull-back of plb-vector bundles 199

6 Examples of projective systems of bundles 207

6.1 Trivial examples of plb-vector bundles 207

6.2 Plb-vector bundles of maps 208

6.3 The inﬁnite jet bundle 211

6.4 The tangent bundle of a plb-bundle 213

6.5 The generalized frame bundle 216

6.6 Generalized associated bundles 219

7 Connections on plb-vector bundles 225

7.1 Projective limits of linear connections 225

7.2 Parallel displacement and holonomy groups 231

7.3 Connections on plb-vector and frame bundles 237

8 Geometry of second order tangent bundles 245

8.1 The (ﬁrst order) tangent bundle in brief 246

8.2 Second order tangent bundles 248

8.3 Second order diﬀerentials 253

8.4 Connection dependence 256

8.5 Second order Fr´echet tangent bundles 257

8.6 Second order frame bundles 262

Appendix: Further study 273

References 281

List of Notations 291

Subject index 299

Preface

The aim of the authors is to lay down the foundations of the pro jective

systems of various geometrical structures modelled on Banach spaces,

eventually leading to homologous structures in the framework of Fr´echet

diﬀerential geometry, by overcoming some of the inherent deﬁciencies of

Fr´echet spaces. We elaborate this brief description in the sequel.

Banach spaces, combining a metric topology (subordinate to a norm),

and a linear space structure (for representing derivatives as linear ap-

proximations to functions in order to do calculus), provide a very conve-

nient setting for many problems in functional analysis, which we need for

handling calculus on function spaces, usually inﬁnite dimensional. They

are a relatively gentle extension from experience on ﬁnite dimensional

spaces, since many topological properties of spaces and groups of lin-

ear maps, as well as many of the existence and uniqueness theorems for

solutions of diﬀerential equations carry over to the inﬁnite dimensional

case.

Manifolds and ﬁbre bundles modelled on Banach spaces arise from the

synthesis of diﬀerential geometry and functional analysis, thus leading

to important examples of global analysis. Indeed, many spaces of (dif-

ferentiable) maps between appropriate manifolds admit the structure of

Banach manifolds (see, for instance, J. Eells [Eel66,§6]).

On the other hand, as mentioned also in [Eel66], Riemannian mani-

folds, represented as rigid maps on inﬁnite dimensional function spaces,

arise as conﬁguration spaces of dynamical systems, with metrics inter-

preted as kinetic energy. Much of the calculus of variations and Morse

theory is concerned with a function space in diﬀerential geometry—

the Euler-Lagrange operator of a variational problem is interpreted as

a gradient vector ﬁeld, with integral curves the paths of steepest as-

cent. Some eigenvalue problems in integral and diﬀerential equations are

vii

viii Preface

interpretable via Lagrangian multipliers, involving inﬁnite dimensional

function spaces from diﬀerential geometry—such as focal point theory

and geometric consequences of the inverse function theorem in inﬁnite

dimensions.

However, in a number of situations that have signiﬁcance in global

analysis and physics, for example, physical ﬁeld theory, Banach space

representations break down. A ﬁrst step forward is achieved by weaken-

ing the topological requirements: Instead of a norm, a family of semi-

norms is considered. This leads to Fr´echet spaces, which do have a linear

structure and their topology is deﬁned through a sequence of seminorms.

Although Fr´echet spaces seem to be very close to Banach spaces, a

number of critical deﬁciencies emerge in their framework. For instance,

despite the progress in particular cases, they lack a general solvability

theory of diﬀerential equations, even the linear ones; also, the space of

continuous linear morphisms between Fr´echet spaces does not remain

in the category, and the space of linear isomorphisms does not admit a

reasonable Lie group structure.

The situation becomes much more complicated when we consider man-

ifolds modelled on Fr´echet spaces. Fundamental tools such as the expo-

nential map of a Fr´echet-Lie group may not exist. Additional compli-

cations become particularly noticeable when we try to collect Fr´echet

spaces together to form bundles (over manifolds modelled on atlases of

Fr´echet spaces), in order to develop geometrical operators like covariant

derivatives and curvature to act on sections of bundles. The structure

group of such bundles, being the general linear group of a Fr´echet space,

is not a Lie group—even worse, it does not have a natural topological

structure. Parallel translations do not necessarily exist because of the in-

herent diﬃculties in solving diﬀerential equations within this framework,

and so on.

This has relevance to real problems. The space of smooth functions

C∞(I, R), where Iis a compact interval of R, is a Fr´echet space. The

space C∞(M, V ), of smooth sections of a vector bundle Vover a com-

pact smooth Riemannian manifold Mwith covariant derivative ∇, is a

Fr´echet space. The C∞Riemannian metrics on a ﬁxed closed ﬁnite-

dimensional orientable manifold has a Fr´echet model space. Fr´echet

spaces of sections arise naturally as conﬁgurations of a physical ﬁeld.

Then the moduli space, consisting of inequivalent conﬁgurations of the

physical ﬁeld, is the quotient of the inﬁnite-dimensional conﬁguration

space Xby the appropriate symmetry gauge group. Typically, Xis

Preface ix

modelled on a Fr´echet space of smooth sections of a vector bundle over

a closed manifold.

Despite their apparent diﬀerences, the categories of Banach and Fr´e-

chet spaces are connected through pro jective limits. Indeed, the limiting

real product space R∞= limn→∞ Rnis the simplest example of this sit-

uation. Taking notice of how R∞arises from Rn, this approach extends

to arbitrary Fr´echet spaces, since always they can be represented by a

countable sequence of Banach spaces in a somewhat similar manner. Al-

though careful concentration to the above example is salutary, (bringing

to mind the story of the mathematician drafted to work on a strate-

gic radar project some 70 years ago, who when told of the context said

“but I only know Ohms Law!” and the response came, “you only need

to know Ohms Law, but you must know it very, very well”), it should

be emphasized that the mere properties of R∞do not answer all the

questions and problems referring to the more complicated geometrical

structures mentioned above.

The approach adopted is designed to investigate, in a systematic way,

the extent to which the shortcomings of the Fr´echet context can be

worked round by viewing, under suﬃcient conditions, geometrical ob-

jects and properties in this context as limits of sequences of their Ba-

nach counterparts, thus exploiting the well developed geometrical tools

of the latter. In this respect, we propose, among other generalizations,

the replacement of certain pathological structures and spaces such as

the structural group of a Fr´echet bundle, various spaces of linear maps,

frame bundles, connections on principal and vector bundles etc., by ap-

propriate entities, susceptible to the limit process. This extends many

classical results to our framework and, to a certain degree, bypasses its

drawbacks.

Apart from the problem of solving diﬀerential equations, much of our

work is motivated also by the need to endow inﬁnite-dimensional Lie

groups with an exponential map [a fact characterizing–axiomatically–

the category of (inﬁnite-dimensional) regular Lie groups]; the diﬀerential

and vector bundle structure of the set of inﬁnite jets of sections of a Ba-

nach vector bundle (compare with the diﬀerential structure described in

[Tak79]); the need to put in a wider perspective particular cases of pro-

jective limits of manifolds and Lie groups appearing in physics (see e.g.

[AM99], [AI92], [AL94], [Bae93]) or in various groups of diﬀeomorphisms

(e.g. [Les67], [Omo70]).

For the convenience of the reader, we give an outline of the presen-

xPreface

tation, referring for more details to the table of contents and the intro-

duction to each chapter.

Chapter 1introduces the basic notions and results on Banach manifolds

and bundles, with special emphasis on their geometry. Since there is not

a systematic treatment of the general theory of connections on Banach

principal and vector bundles (apart from numerous papers, with some

very fundamental ones among them), occasionally we include extra de-

tails on speciﬁc topics, according to the needs of subsequent chapters.

With a few exceptions, there are not proofs in this chapter and the

reader is guided to the literature for details. This is to keep the notes

within a reasonable size; however, the subsequent chapters are essentially

self-contained.

Chapter 2contains a brief account of the structure of Fr´echet spaces and

the diﬀerentiability method applied therein. From various possible diﬀer-

entiability methods we have chosen to apply that of J.A. Leslie [Les67],

[Les68], a particular case of Gˆateaux diﬀerentiation which ﬁts well to the

structure of locally convex spaces, without recourse to other topologies.

Among the main features of this chapter we mention the representation

of a Fr´echet space by a projective limit of Banach spaces, and that of

some particular spaces of continuous linear maps by projective limits of

Banach functional spaces, a fact not true for arbitrary spaces of linear

maps. An application of the same representation is proposed for study-

ing diﬀerential equations in Fr´echet spaces, including also comments on

other approaches to the same subject. Projective limit representations

of various geometrical structures constitute one of the main tools of our

approach.

Chapter 3is dealing with the smooth structure, under appropriate con-

ditions, of Fr´echet manifolds arising as projective limits of Banach man-

ifolds, as well as with topics related to their tangent bundles. The case of

Fr´echet-Lie groups represented by projective limits of Banach-Lie groups

is also studied in detail, because of their fundamental role in the struc-

ture of Fr´echet principal bundles. Such groups admit an exponential

map, an important property not yet established for arbitrary Fr´echet-

Lie groups.

Chapter 4is devoted to the study of projective systems of Banach prin-

cipal bundles and their connections. The latter are handled by their

connection forms, global and local ones. It is worthy of note that any

Fr´echet principal bundle, with structure group one of those alluded to

in Chapter 3, is always representable as a pro jective limit of Banach

Preface xi

principal bundles, while any connection on the former bundle is an ap-

propriate projective limit of connections in the factor bundles of the

limit. Here, related (or conjugate) connections, already treated in Chap-

ter 1, provide an indispensable tool in the approach to connections in

the Fr´echet framework. We further note that the holonomy groups of

the limit bundle do not necessarily coincide with the projective limits

of the holonomy groups of the factor bundles. This is supported by an

example after the study of ﬂat bundles.

Chapter 5is concerned with projective limits of Banach vector bundles.

If the ﬁbre type of a limit bundle is the Fr´echet space F, the structure of

the vector bundle is fully determined by a particular group (denoted by

H0(F) and described in §5.1), which replaces the pathological general

linear group GL(F) of F, thus providing the limit with the structure of

a Fr´echet vector bundle. The study of connections on vector bundles of

the present type is deferred until Chapter 7.

Chapter 6contains a collection of examples of Fr´echet bundles realized

as projective limits of Banach ones. Among them, we cite in particular

the bundle J∞(E) of inﬁnite jets of sections of a Banach vector bundle

E. This is a non trivial example of a Fr´echet vector bundle, essentially

motivating the conditions required to deﬁne the structure of an arbi-

trary vector bundle in the setting of Chapter 5. On the other hand, the

generalized bundle of frames of a Fr´echet vector bundle is an important

example of a principal bundle with structure group the aforementioned

group H0(F).

Chapter 7aims at the study of connections on Fr´echet vector bundles the

latter being in the sense of Chapter 5. The relevant notions of parallel

displacement along a curve and the holonomy group are also examined.

Both can be deﬁned, despite the inherent diﬃculties of solving equations

in Fr´echet spaces, by reducing the equations involved to their counter-

parts in the factor Banach bundles.

Chapter 8is mainly focused on the vector bundle structure of the second

order tangent bundle of a Banach manifold. Such a structure is always

deﬁned once we choose a linear connection on the base manifold, thus

a natural question is to investigate the dependence of the vector bun-

dle structure on the choice of the connection. The answer relies on the

possibility to characterize the second order diﬀerentials as vector bundle

morphisms, which is aﬃrmative if the connections involved are properly

related (conjugate). The remaining part of the chapter is essentially an

xii Preface

application of our methods to the second order Fr´echet tangent bundle

and the corresponding (generalized) frame bundle.

We conclude with a series of open problems or suggestions for further

applications, within the general framework of our approach to Fr´echet

geometry, eventually leading to certain topics not covered here.

These notes are addressed to researchers and graduate students of math-

ematics and physics with an interest in inﬁnite-dimensional geometry,

especially that of Banach and Fr´echet manifolds and bundles. Since we

have in mind a wide audience, with possibly diﬀerent backgrounds and

interests, we have paid particular attention to the details of the exposi-

tion so that it is as far as possible self-contained. However, a familiarity

with the rudiments of the geometry of manifolds and bundles (at least

of ﬁnite dimensions) is desirable if not necessary.

It is a pleasure to acknowledge our happy collaboration, started over

ten years ago by discussing some questions of common research interest

and resulting in a number of joint papers. The writing of these notes is

the outcome of this enjoyable activity. Finally, we are very grateful to

an extremely diligent reviewer who provided many valuable comments

and suggestions on an earlier draft, we have beneﬁted much from this in

the ﬁnal form of the monograph.

Manchester – Piraeus – Athens,

February 2015

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