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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