Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry Volume II: Geometry. Examples and Applications

456 Pages·1998·17.623 MB·English

by  Anastasios Mallios (auth.)

Geometry ofVector Sheaves
Mathematics and Its Applications
ManagingEditor:
M.HAZEWINKEL
Centre/orMathematicsandComputerScience.Amsterdam.TheNetherlands
Volume439
Geometry of
Vector Sheaves
An Axiomatic Approach
to Differential Geometry
Volume 11: Geometry. Examples and Applications
by
Anastasios Mallios
Department 0/ Mathematics,
University 0/A thens,
Athens, Greece
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. CataIogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6102-5 ISBN 978-94-011-5006-4 (eBook)
DOI 10.1007/978-94-011-5006-4
Printed on acid-free paper
All Rights Reserved
© 1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998
Softcover reprint ofthe hardcover 1st edition 1998
No part of the material protected by this copyright notice may be reproduced or
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Contents
General Preface (to both Volumes I, II) , ix
Preface (to Volume II) xiii
Acknowledgements . .... ....... ..... .. ... .............. ..... .... xix
Contents of Volume I xxi
Part Two. Geometry
CHAPTER VI. Geometry of Vector Sheaves. A-connections
1. C-derivations. Differential triads 2
1.(a). Logarithmic derivation 6
,
x ni)
2. A standard example: The smooth triad (C d, 9
3. A-connections. Basic definitions 10
4. Examples of A-connections 13
4.(a). The standard (flat) A-connection 13
4.(b). Cx-connections 14
5. Induced A-connections , 16
5.(a). Whitney sum of A-connections 16
5.(b). Tensor product ofA-connections 18
5.(c). A-connections vis-a-vis to the functor HamA 19
5.(d). Dual A-connections 22
6. Induced A-connections (contn'd). Pull-back of A-connections 24
6.(a). Restriction ofA-connections 28
7. Affine space of A-connections 29
8. Localization of A-connections 36
8.(a). Local Levi-Civita A-connections 41
9. Sheafof connection coefficients , 44
10. Levi-Civita l-cocycles, as related to coordinate l-cocycles
of vector sheaves 48
11. Existence of A-connections 53
11.(a). Integrable A-connections 56
11.(b). Levi-Civita A-connections of An, n E N, 57
12. A-connections as splittings ofjet-line sheaves 62
13. A-connections as splitting extensions (contn'd) 68
14. Coordinate l-cocycles of A-connections, as related to A-connections 71
v
15. Examples (contn'd) 74
15.(a). Coo-manifolds 75
15.(b). Complex manifolds. Holomorphic connections 76
16. Fine vector sheaves '" 82
17. Moduli space of A-connections 86
18. Cohomological classification ofline sheaves admitting
A-connections 92
Chapter VII. A-connections. Local Theory
1. Local form of A-connections 98
2. Change oflocal gauges 106
2.(a). Vector sheaves 111
3. Transformation law of potentials. Existence of A-connections 116
4. Dual A-connections. Local form 119
5. Christoffel sections 123
6. Complete parallelism " 136
7. a-flat line sheaves. Generalized Selesnick-Chern class of a
line sheaf 144
7.(a). Complex Chern class ofa line sheaf (contn'd) 150
8. Symmetric A-connections 153
9. Riemannian vector sheaves. Fundamental lemma (general case) 167
10. Hermitian A-connections ,..171
11. Cohomological classification of Maxwell fields (contn'd) 174
CHAPTER VIII. Curvature
1. Preliminaries. Basic Definitions 186
2. Curvature of an A-connection 190
3. Local form of the curvature. Cartan's structural equation 194
4. Change oflocal gauge 198
4.(a). Adjoint representation of a sheafofgroups in a
given A-module 198
4.(b). Change of curvature under a gauge transformation 200
5. Flat A-connections (contn'd) 203
5.(a). Frobenius integrability condition 205
VI
6. Trivial A-connections 214
7. Bianchi's identity 219
8. Higher exterior derivations and prolongations 226
8.(a). Trace ofcurvature 230
9. Curvatures ofinduced A-connections 231
10. Torsion of an A-connection. Local form 236
11. Weil's integrality theorem 238
CHAPTER IX. Characteristic Classes
1. Preliminaries 244
2. Invariant polynomials 247
3. Generalized de Rham spaces. Fundamental theorem 254
4. Symmetric polynomials 263
5. Generalized Chern classes 266
6. Chern-WeiIhomomorphism 271
PartThree. Examples and Applications
CHAPTER X. Classical Theory
1. Differential geometry ofCoo-manifolds. (Finite and
infinite-dimensional case) 278
l.(a). Infinite-dimensional example 281
2. Holomorphy 285
3. Generalized manifolds (contn'd) 286
3.(a). Orbifolds (or Satake manifolds) 288
4. Electromagnetism 289
5. Elementary particles 290
6. Supermanifolds 294
7. Some more applications 298
CHAPTER XI. Sheaves and Presheaves with Topological
Algebraic Structures
1. Preliminaries. Topological algebra sheaves 300
l.(a). Topological algebra presheaves 300
l.(b). Topological algebra sheaves 301
VII
2. The Gel'fand sheafof a topological algebra 303
3. Geometric topological algebras 311
4. Softness of the Gel'fand sheaf 317
5. De Rham-Kiihler complex of a topological algebra 321
6. Sheafification of the de Rham-Kiihler complex. Logarithmic
derivation 326
6.(a). Logarithmic derivation 329
7. The sheafexponential sequence 330
8. Line sheaves over the spectrum ofa geometric topological
algebra 338
8.(a). Morphisms of short exact exponential sheaf
sequences (contn'd) 348
9. Frobenius integrability condition (contn'd) 353
10. Flat A-vector bundles vis-a-vis to a-flatness of the
associated vector sheaves 359
10.(a). a-flatness of A-vector bundles 361
11. Watts-Selesnick complex 363
11.(a). Sheafification of the Watts-Selesnick complex 367
12. Further applications 376
12.(a). Fourier algebras 377
12.(b). Formal power series algebras 382
BIBLIOGRAPHY 387
Notational Index 401
Subject Index 425
VIII
General Preface (to both Volumes I, II)
Our aim by this study is to exhibit and also exploit a quite general tech
nique, yet simple, by its very nature, which enables one to formulate, at
least, a substantial amount (if not all!) of the fundamental notions of the
classical differential geometry of Coo-manifolds and obtain too several stan
dard results thereof. More interestingly, this can be achieved (we might say
here, surprisingly enough!), by no use at all (!) of any concept of tangent
vectors or of differential forms, in the classical sense of these words; indeed,
no calculus is employedaltogether! This is, generally speaking, our program,
through the present two-volume work, whose secondary title to the adopted
main one might be, of course, An Introduction to (an) Abstract (treatise of)
Differential Geometry. Such a perspective, even if subconciously evident,
more or less, to the experts, has never been, however, quite explicit, to the
extent and generality, at least, which is undertaken by this discussion, so
that the popularization ofthis point ofview might be ofsomeinterest, much
more because, as we shall see, there do exist considerable possibilities for
applications.
Treatments aiming at the exposition ofseveral aspects of differential ge
ometry, outsidetheclassicalframework of(finite-dimensional)COO-manifolds,
range, of course, from the standard theory of infinite-dimensional Coo-mani
folds, with a vast already literature thereon, to the more sophisticated as
pects of "differential geometry" on topological spaces, more general than
Coo-manifolds. However, this has always been done, through a suitable for
mulation of the basic notions of the standard case, as for instance, vector
fields, differential forms and the like.
By contrast, our treatment is quite general and, so to speak, axiomatic.
So the basic tools here are sheaves ofmodules, with respect to an appropriate
sheafofC-algebras (alias, C-algebra shea!), our "domain ofcoefficients", by
analogy with the classical case. Yet, the sheaves involved are mostlyover an
arbitrary topological space X, that eventually is decreed to be paracompact
(Hausdorff). The latter requirement for X, base space of the sheaves con
sidered, is only entered into the discussion, to profit from it cohomologically,
just as occasion serves; so this is, for instance, the case with the counterpart
here of Weil's integrality theorem, pertaining, as we know, to the cohomol-
IX
ogy class, determined by the curvature form of a connection of a given line
bundle (sheaf).
Thus, sheafcohomologyis the second basic constituent ofour armament,
in order to carry out a rather ambitious(!) program, as is namely, the trans
fer, to the present abstract setting, ofseveralfundamental results, ofa global
and/or local nature, of differential geometry of manifolds. Indeed, it is, at
least, interesting (and also encouraging!, for an undertaking, like the one at
hand) to realize how a considerable amount of the classical theory can be
carried over to the present formalism, especially in what concerns an im
portant differential geometric aspect, as is the theory of connections and its
consequences, as e.g. curvature, characteristic classes (it la Chern-WeiI) and
so on.
Our study is, as already mentioned, quite general and essentially of an
algebraic (topological) nature, based, as actually does, on sheaftheory. So it
contains, as a very special case, to theextent, ofcourse, that is accomplished
herewith, the classical aspect of finite (and, partly, of infinite) dimensional
differential geometry of Coo-manifolds, as well as, several other generaliza
tions thereon, as alluded to at the beginning of this Preface. The latter
cases are also discussed, as particular non-standard examples, at the perti
nent places in the sequel (see thus Volume II, Part Three).
In this context, another item worth pointing out here is the crucial role
played, in a number of places, by the corresponding to our case de Rham
complex of the standard theory. This refers, in particular, to the exactness
of that complex (here deemed to be in force), in the classical case being, of
course, a consequence ofthe famous Poincare Lemma (viz. that every closed
form is exact). Yet, in the aforementioned (non-standard) examples exact
ness ofthe previous complexis still valid, as a consequence again ofthe same
lemma, which thus is proved to hold true in those extended cases, as well.
On the other hand, in our abstract setting the situation can be remedied by
demanding, for instance, the section algebras of the C-algebra sheaf A, our
"domain of coefficients", to carry further a topological algebraic structure,
being, thus, in effect, appropriate topological algebras. So our "sheafofcoef
ficients" becomes now, precisely speaking, a topological C-algebra sheaf on
X.
The insert at this place of topological algebra theory is both crucial and
decisive. Of course, this can be construed as revealing, within the present
general framework, the oblique, nontheless, significant role, played in the
x