Progress in Mathematical Physics

By: Bozhidar Z. Iliev

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Hardcover | 15 September 2006

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The main subject of this book is an up-to-date and in-depth survey of the theory of normal frames and coordinates in di?erential geometry. The existing results, as well as new ones obtained lately by the author, on the theme are presented. The text is so organized that it can serve equally well as a reference manual, introduction to and review of the current research on the topic. Correspondingly, the possible audience ranges from graduate and post-graduate students to sci- tists working in di?erential geometry and theoretical/mathematical physics. This is re?ected in the bibliography which consists mainly of standard (text)books and journal articles. The present monograph is the ?rst attempt for collecting the known facts concerting normal frames and coordinates into a single publication. For that r- son, the considerations and most of the proofs are given in details. Conventionally local coordinates or frames, which can be holonomic or not, are called normal if in them the coe?cients of a linear connection vanish on some subset, usually a submanifold, of a di?erentiable manifold. Until recently the ex- tence of normal frames was known (proved) only for symmetric linear connections on submanifolds of a manifold. Now the problems concerning normal frames for derivationsof thetensor algebraovera di?erentiablemanifoldarewellinvestigate; in particular they completely cover the exploration of normal frames for arbitrary linear connections on a manifold. These rigorous results are important in conn- tion with some physical applications.

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"This is a careful and rather comprehensive presentation of theory and methods related to normal frames in differential geometry. ... The book is carefully written, provides a good overview, and contains many interesting aspects not available otherwise. It is a valuable addition to the literature on connections in differential geometry." (M. Kunzinger, Monatshefte fuer Mathematik, Vol. 154 (1), May, 2008)

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry

List of Conventions	p. xi
Preface	p. xiii
Acknowledgments	p. xvi
Manifolds, Normal Frames and Riemannian Coordinates
Introduction	p. 2
Differentiable manifolds	p. 4
Basic definitions	p. 4
Differentiable mappings	p. 7
Tangent vectors and vector fields	p. 9
Covectors and covector fields	p. 15
Tensors and tensor fields. Tensor algebras	p. 17
Linear connections on a differentiable manifold	p. 19
Motivation	p. 20
Basic definitions	p. 21
Parallel transport	p. 26
Geodesics and exponential mapping	p. 30
Riemannian manifolds	p. 33
Normal frames: general ideas, uniqueness and holonomicity	p. 37
Normal coordinates on Riemannian manifolds	p. 44
Examples of normal coordinates for Riemannian connections	p. 49
Terminology 1: Bases and frames. Holonomicity	p. 68
Conclusion	p. 71
Existence, Uniqueness and Construction of Normal Frames and Coordinates for Linear Connections
Introduction	p. 74
The case at a single point	p. 75
Old classical method	p. 75
Complete description	p. 79
Modern classical method	p. 83
The case along paths without self-intersections	p. 84
Fermi coordinates	p. 85
Complete description	p. 93
The case in a neighborhood	p. 104
The case on arbitrary submanifolds	p. 111
Conventional method	p. 112
Complete description	p. 120
Examples of normal frames and coordinates	p. 128
Conclusion	p. 137
Normal Frames and Coordinates for Derivations on Differentiable Manifolds
Introduction	p. 142
Derivations of the tensor algebra over a manifold	p. 143
General overview	p. 148
Frames and coordinates normal at a point	p. 153
Frames and coordinates normal along paths	p. 154
Frames and coordinates normal in a neighborhood	p. 157
Frames and coordinates normal on submanifolds	p. 158
Frames and coordinates normal along mappings	p. 161
Injective mappings	p. 162
Locally injective mappings	p. 168
Mappings between manifolds	p. 169
Normal frames and coordinates for derivations along a fixed vector field	p. 172
The case at a single point	p. 175
The case along paths	p. 178
The case on the whole manifold	p. 184
Other cases	p. 186
Normal frames and coordinates for derivations along paths	p. 189
On frames simultaneously normal for two derivations	p. 197
Normal frames for linear connections (review)	p. 202
Examples	p. 204
Terminology 2: Normal and geodesic frames	p. 209
Conclusion	p. 210
Normal Frames in Vector Bundles
Introduction	p. 214
Vector bundles	p. 216
Basic definitions	p. 217
Liftings of paths	p. 218
Derivations along paths	p. 219
Tensor bundles	p. 222
Linear transports along paths in vector bundles	p. 225
Definition and general form	p. 225
Representations in frames along paths	p. 228
Linear transports and derivations along paths	p. 230
Normal frames for linear transports	p. 234
On the existence of normal frames	p. 239
The case of a manifold as a base	p. 247
Linear transports and normal frames in line bundles	p. 254
Normal frames for derivations in vector bundles with a manifold as a base	p. 257
Curvature and normal frames	p. 261
Curvature of linear transport or derivation along paths	p. 261
On the curvature of Euclidean transports along paths	p. 268
Torsion and normal coordinates	p. 271
Torsion of linear transport or derivation along paths in the tangent bundle over a manifold	p. 271
Holonomic normal frames in the tangent bundle	p. 276
Parallel transports in tangent bundles	p. 280
The parallel transport as a transport along paths	p. 281
Normal frames for parallel transports along paths	p. 285
Strong normal frames	p. 289
Linear transports assigned to derivations in tangent bundles	p. 292
Derivations along paths	p. 293
Derivations along vector fields	p. 295
Derivations along fixed vector field	p. 295
Normal frames	p. 298
Links with the theory of connections and parallel transports	p. 301
Parallelism structures, connections and covariant derivatives	p. 301
Parallel transports in vector bundles	p. 311
Parallel transports and linear transports along paths	p. 315
Normal frames for parallel transports, connections and covariant derivatives	p. 320
On the role of the curvature	p. 323
Autoparallel paths	p. 325
On a fibre bundle view at quantum mechanics	p. 328
Conclusion	p. 330
Normal Frames for Connections on Differentiable Fibre Bundles
Introduction	p. 334
Preliminaries	p. 336
Connections on bundles	p. 339
Frames and coframes on the bundle space	p. 339
Connection theory	p. 342
Connections on vector bundles	p. 350
Vertical lifts	p. 350
Linear connections on vector bundles	p. 353
Covariant derivatives in vector bundles	p. 357
Affine connections	p. 361
General (co)frames	p. 365
Normal frames	p. 375
The general case	p. 375
Normal frames adapted to holonomic frames	p. 377
Normal frames on vector bundles	p. 381
Coordinates normal along injective mappings with non-vanishing horizontal component	p. 384
Links between connections and transports along paths in fibre bundles	p. 89
Conclusion	p. 398
Bibliography	p. 401
Subject List of Symbols	p. 411
Set theory	p. 411
Linear/vector spaces	p. 412
Matrices	p. 413
Manifolds	p. 414
Linear connections on manifolds	p. 416
Derivations on manifolds	p. 416
Fibre bundles	p. 417
Linear transports and derivations along paths	p. 418
Parallel transports, connections and covariant derivatives	p. 419
Auhtor Index	p. 421
Notation Index	p. 423
Subject Index	p. 429
Table of Contents provided by Ingram. All Rights Reserved.

Progress in Mathematical Physics

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Hardcover

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