CONTENTS v
Notes and commentary 251
Chapter 9. Type I singularities 253
1. Intuition 253
2. Positive curvature is preserved 255
3. Positive sectional curvature dominates 256
4. Necklike points in finite-time singularities 262
5. Necklike points in ancient solutions 271
6. Type I ancient solutions on surfaces 274
Notes and commentary 277
Appendix A. The Ricci calculus 279
1. Component representations of tensor fields 279
2. First-order differential operators on tensors 280
3. First-order differential operators on forms 283
4. Second-order differential operators 284
5. Notation for higher derivatives 285
6. Commuting covariant derivatives 286
Notes and commentary 286
Appendix B. Some results in comparison geometry 287
1. Some results in local geometry 287
2. Distinguishing between local geometry and global geometry 295
3. Busemann functions 303
4. Estimating injectivity radius in positive curvature 312
Notes and commentary 315
Bibliography 317
Index
323
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