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Contents
Preface vii
Chapter l. Riemannian Metrics 1
1. Riemannian Manifolds and Maps 2
2. Groups and Riemannian Manifolds 5
3. Local Representations of Metrics 8
4. Doubly Warped Products 13
5. Exercises 17
Chapter 2. Curvature 21
1. Connections 22
2. The Connection in Local Coordinates 29
3. Curvature 32
4. The Fundamental Curvature Equations 41
5. The Equations of Riemannian Geometry 47
6. Some Tensor Concepts 51
7. Further Study 56
8. Exercises 56
Chapter 3. Examples 63
1. Computational Simplifications 63
2. Warped Products 64
3. Hyperbolic Space 74
4. Metrics on Lie Groups 77
5. Riemannian Submersions 82
6. Further Study 90
7. Exercises 90
Chapter 4. Hypersurfaces 95
1. The Gauss Map 95
2. Existence of Hypersurfaces 97
3. The Gauss-Bonnet Theorem 101
4. Further Study 107
5. Exercises 108
Chapter 5. Geodesics and Distance 111
1. Mixed Partials 112
2. Geodesics 116
3. The Metric Structure of a Riemannian Manifold 121
4. First Variation of Energy 126
5. The Exponential Map 130
6. Why Short Geodesics Are Segments 132
7. Local Geometry in Constant Curvature 134
8. Completeness 137
9. Characterization of Segments 139
10. Riemannian Isometries 143
11. Further Study 149
12. Exercises 149
Chapter 6. Sectional Curvature Comparison I 153
1. The Connection Along Curves 153
2. Second Variation of Energy 158
3. Nonpositive Sectional Curvature 162
4. Positive Curvature 169
5. Basic Comparison Estimates 173
6. More on Positive Curvature 176
7. Further Study 182
8. Exercises 183
Chapter 7. The Bochner Technique 187
1. Killing Fields 188
2. Hodge Theory 202
3. Harmonic Forms 205
4. Clifford Multiplication on Forms 213
5. The Curvature Tensor 221
6. Further Study 229
7. Exercises 229
Chapter 8. Symmetric Spaces and Holonomy 235
1. Symmetric Spaces 236
2. Examples of Symmetric Spaces 244
3. Holonomy 252
4. Curvature and Holonomy 256
5. Further Study 262
6. Exercises 263
Chapter 9. Ricci Curvature Comparison 265
1. Volume Comparison 265
2. Fundamental Groups and Ricci Curvature 273
3. Manifolds of Nonnegative Ricci Curvature 279
4. Further Study 290
5. Exercises 290
Chapter 10. Convergence 293
1. Gromov-Hausdorff Convergence 294
2. Holder Spaces and Schauder Estimates 301
3. Norms and Convergence of Manifolds 307
4. Geometric Applications 318
5. Harmonic Norms and Ricci curvature 321
6. Further Study 330
7. Exercises 331
Chapter 11. Sectional Curvature Comparison 11 333
1. Critical Point Theory 333
2. Distance Comparison 338
3. Sphere Theorems 346
4. The Soul Theorem 349
5. Finiteness of Betti Numbers 357
6. Homotopy Finiteness 365
7. Further Study 372
8. Exercises 372
Appendix.De Rham Cohomology 375
1. Lie Derivatives 375
2. Elementary Properties 379
3. Integration of Forms 380
4. Cech Cohomology 383
5. De Rham Cohomology 384
6. Poincare Duality 387
7. Degree Theory 389
8. Further Study 391
Bibliography 393
Index 397
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