Geometry of Manifolds (eBook)

Front Cover 1
Geometry of Manifolds 4
Copyright Page 5
Contents 8
Preface 6
Chapter 1. Manifolds 14
1.1 Introductory Material and Notation 14
1.2 Definition of a Manifold 15
1.3 Tangent Space 20
1.4 Vector Fields 26
1.5 Submanifolds 34
1.6 Distributions and Integrability 35
Chapter 2. Lie Groups 38
2.1 Lie Groups 38
2.2 Lie Algebras 39
2.3 Lie Group—Lie Algebra Correspondence 41
2.4 Homomorphisms 42
2.5 Exponential Map 43
2.6 Representations 47
Chapter 3. Fibre Bundles 51
3.1 Transformation Groups 51
3.2 Principal Bundles 54
3.3 Associated Bundles 58
3.4 Reduction of the Structural Group 62
Chapter 4. Differential Forms 66
4.1 Introduction 66
4.2 Classical Notion of Differential Form 66
4.3 Grassmann Algebras 67
4.4. Existence of Grassmann Algebras 70
4.5 Differential Forms 75
4.6 Exterior Derivative 77
4.7 Action of Maps 81
4.8 Frobenius' Theorem 83
4.9. Vector-Valued Forms and Operations 84
4.10 Forms on Complex Manifolds 85
Chapter 5. Connexions 87
5.1 Definitions and First Properties 87
5.2. Parallel Translation 90
5.3 Curvature Form and the Structural Equation 93
5.4 Existence of Connexions and Connexions in Associated Bundles 96
5.5 Structural Equations for Horizontal Forms 97
5.6 Holonomy 100
Chapter 6. Affine Connexions 102
6.1 Definitions 102
6.2. The Structural Equations of an Affine Connexion 112
6.3 The Exponential Maps 121
6.4 Covariant Differentiation and Classical Forms 124
Chapter 7. Riemannian Manifolds 135
7.1 Definitions and First Properties 135
7.2 The Bundle of Frames 140
7.3 Riemannian Connexions 142
7.4 Examples and Problems 145
Chapter 8. Geodesics and Complete Riemannian ManifoIds 158
8.1 Geodesics 158
8.2 Complete Riemannian Manifolds 165
8.3 Continuous Curves 171
Chapter 9. Riemannian Curvature 174
9.1 Riemannian Curvature 174
9.2 Computation of the Riemannian Curvature 178
9.3 Continuity of the Riemannian Curvature 179
9.4 Rectangles and Jacobi Fields 185
9.5 Theorems Involving Curvature 191
Chapter 10. Immersions and the Second Fundamental Form 198
10.1 Definitions 198
10.2 The Connexions 200
10.3 Curvature 202
10.4 The Second Fundamental Form 203
10.5 Curvature and the Second Fundamental Form 205
10.6 The Local Gauss Map 208
10.7 Hessians of Normal Coordinates of N 210
10.8 A Formulation of the Immersion Problem 212
10.9 Hypersurfaces 220
Chapter 11. Second Variation of Arc Length 226
11.1 First and Second Variation of Arc Length 226
11.2 The Index Form 233
11.3 Focal Points and Conjugate Points 237
11.4 The Infinitesimal Deformations 239
11.5 The Morse Index Theorem 246
11.6 The Minimum Locus 250
11.7 Closed Geodesics 254
11.8 Convex Neighborhoods 259
11.9 Rauch’s Comparison Theorem 263
11.10 Curvature and Volume 266
Appendix. Theorems on Diferential Equations 271
Bibliography 273
Subject Index 278