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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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