CONTENTS vii
Appendix E. Aspects of Geometric Analysis Related to Ricci Flow 379
1. Green's function 379
2. Positive and fundamental solutions to the heat equation 388
3. Li-Yau differential Harnack estimate 403
4. Gradient estimates for the heat equation 405
Appendix F. Tensor Calculus on the Frame Bundle 411
1. Introduction 411
2. Tensors as vector-valued functions on the frame bundle 412
3. Local coordinates on the frame bundle 414
4. The metric on the frame bundle 415
5. A natural frame field on FM 416
6. Covariant differentiation 420
7. Curvature and commuting covariant derivatives 422
8. Reduction to the orthonormal frame bundle 423
9. Time dependent orthonormal frame bundle for solutions to the
Ricci flow 426
10. The time vector field and its action on tensors 427
11. The heat operator and commutation formulas 429
12. Notes and commentary 431
Bibliography 433
Index 455
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