Author(s): Dmitri Burago, Yuri Burago and Sergei Ivanov
Series: Graduate Studies in Mathematics 33
Publisher: American Mathematical Society
Year: 2001
Language: English
Pages: 426
Preface vii
Chapter 1. Metric Spaces 1
x1.1. De¯nitions 1
x1.2. Examples 3
x1.3. Metrics and Topology 7
x1.4. Lipschitz Maps 9
x1.5. Complete Spaces 10
x1.6. Compact Spaces 13
x1.7. Hausdor® Measure and Dimension 17
Chapter 2. Length Spaces 25
x2.1. Length Structures 25
x2.2. First Examples of Length Structures 30
x2.3. Length Structures Induced by Metrics 33
x2.4. Characterization of Intrinsic Metrics 38
x2.5. Shortest Paths 44
x2.6. Length and Hausdor® Measure 53
x2.7. Length and Lipschitz Speed 55
Chapter 3. Constructions 59
x3.1. Locality, Gluing and Maximal Metrics 59
x3.2. Polyhedral Spaces 67
x3.3. Isometries and Quotients 74
x3.4. Local Isometries and Coverings 78
x3.5. Arcwise Isometries 85
x3.6. Products and Cones 87
Chapter 4. Spaces of Bounded Curvature 101
x4.1. De¯nitions 101
x4.2. Examples 109
x4.3. Angles in Alexandrov Spaces and Equivalence of De¯nitions 114
x4.4. Analysis of Distance Functions 119
x4.5. The First Variation Formula 121
x4.6. Nonzero Curvature Bounds and Globalization 126
x4.7. Curvature of Cones 131
Chapter 5. Smooth Length Structures 135
x5.1. Riemannian Length Structures 136
x5.2. Exponential Map 150
x5.3. Hyperbolic Plane 154
x5.4. Sub-Riemannian Metric Structures 178
x5.5. Riemannian and Finsler Volumes 193
x5.6. Besikovitch Inequality 202
Chapter 6. Curvature of Riemannian Metrics 209
x6.1. Motivation: Coordinate Computations 211
x6.2. Covariant Derivative 214
x6.3. Geodesic and Gaussian Curvatures 221
x6.4. Geometric Meaning of Gaussian Curvature 226
x6.5. Comparison Theorems 237
Chapter 7. Space of Metric Spaces 241
x7.1. Examples 242
x7.2. Lipschitz Distance 249
x7.3. Gromov{Hausdor® Distance 251
x7.4. Gromov{Hausdor® Convergence 260
x7.5. Convergence of Length Spaces 265
Chapter 8. Large-scale Geometry 271
x8.1. Noncompact Gromov{Hausdor® Limits 271
x8.2. Tangent and Asymptotic Cones 275
x8.3. Quasi-isometries 277
x8.4. Gromov Hyperbolic Spaces 284
x8.5. Periodic Metrics 298
Chapter 9. Spaces of Curvature Bounded Above 307
x9.1. De¯nitions and Local Properties 308
x9.2. Hadamard Spaces 324
x9.3. Fundamental Group of a Nonpositively Curved Space 338
x9.4. Example: Semi-dispersing Billiards 341
Chapter 10. Spaces of Curvature Bounded Below 351
x10.1. One More De¯nition 352
x10.2. Constructions and Examples 354
x10.3. Toponogov's Theorem 360
x10.4. Curvature and Diameter 364
x10.5. Splitting Theorem 366
x10.6. Dimension and Volume 369
x10.7. Gromov{Hausdor® Limits 376
x10.8. Local Properties 378
x10.9. Spaces of Directions and Tangent Cones 390
x10.10. Further Information 398
Bibliography 405
Index 409
