Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow.Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models. The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow. The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.
Author(s): Keith Burns, Marian Gidea
Series: Studies in Advanced Mathematics
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2005
Language: English
Pages: 400
Tags: Математика;Топология;Дифференциальная геометрия и топология;
Front cover......Page 1
Preface......Page 9
Contents......Page 10
1.1 Introduction......Page 13
1.2 Review of topological concepts......Page 16
1.3 Smooth manifolds......Page 21
1.4 Smooth maps......Page 28
1.5 Tangent vectors and the tangent bundle......Page 31
1.6 Tangent vectors as deriv ations......Page 39
1.7 The derivative of a smooth map......Page 42
1.8 Orientation......Page 45
1.9 Immersions, embeddings and submersions......Page 48
1.10 Regular and critical points and values......Page 57
1.11 Manifolds with boundary......Page 60
1.12 Sard's theorem......Page 65
1.13 Transversality......Page 71
1.14 Stability......Page 74
1.15 Exercises......Page 78
2.1 Introduction......Page 83
2.2 Vector fields......Page 86
2.3 Differential equations and smooth dynamical systems......Page 92
2.4 Lie derivative, Lie bracket......Page 98
2.5 Discrete dynamical systems......Page 106
2.6 Hyperbolic fixed points and periodic orbits......Page 109
2.7 Exercises......Page 118
3.1 Introduction......Page 121
3.2 Riemannian metrics and the first fundamental form......Page 124
3.3 Standard geometries on surfaces......Page 133
3.4 Exercises......Page 137
4.1 Introduction......Page 139
4.2 Affine connections......Page 143
4.3 Riemannian connections......Page 148
4.4 Geodesics......Page 154
4.5 The exponential map......Page 161
4.6 Minimizing properties of geodesics......Page 167
4.7 The Riemannian distance......Page 174
4.8 Exercises......Page 179
5.1 Introduction......Page 183
5.2 The curvature tensor......Page 188
5.3 The second fundamental form......Page 196
5.4 Sectional and Ricci curvatures......Page 207
5.5 Jacobi fields......Page 213
5.6 Jacobi fields on manifolds of constant curvature......Page 220
5.7 Conjugate points......Page 222
5.8 Horizontal and vertical sub-bundles......Page 225
5.9 The geodesic flow......Page 229
5.10 Exercises......Page 234
6.1 Introduction......Page 237
6.2 Vector bundles......Page 239
6.3 The tubular neighborhood theorem......Page 243
6.4 Tensor bundles......Page 245
6.5 Differential forms......Page 250
6.6 Integration of differential forms......Page 259
6.7 Stokes' theorem......Page 263
6.8 De Rham cohomology......Page 269
6.9 Singular homology......Page 275
6.10 The de Rham theorem......Page 283
6.11 Exercises......Page 288
7.1 Introduction......Page 291
7.2 The Brouwer degree......Page 294
7.3 The oriented intersection number......Page 303
7.4 The fixed point index and the index of a vector field......Page 305
7.5 The Lefschetz number......Page 315
7.6 The Euler characteristic......Page 318
7.7 The Gauss-Bonnet theorem......Page 325
7.8 Exercises......Page 336
8.1 Introduction......Page 339
8.2 Functions with nondegenerate critical points......Page 341
8.3 The gradient flow......Page 349
8.4 The topology of level sets......Page 352
8.5 Manifolds represented as CW complexes......Page 360
8.6 Morse inequalities......Page 363
8.7 Exercises......Page 368
9.1 Introduction......Page 369
9.2 Hyperbolic sets......Page 371
9.3 Hyperbolicity criteria......Page 380
9.4 Geodesic flows on compact Riemannian manifolds with negative sectional curvature......Page 385
9.5 Exercises......Page 388
References......Page 391
Index......Page 397
Back cover......Page 403
