Acknowledgments ............................................... vii
1 Introduction ................................................. 1
1.1 Related questions ....................................... 8
1.2 Outline of this work ................................... 11
2 An Iterative Decomposition of Global Conformal Invariants:
The First Step .............................................. 19
2.1 Introduction ........................................... 19
2.2 Conventions, background, and the super divergence
formula ................................................ 27
2.3 From the super divergence formula for Ig(Ø) back to
P(g): The two main claims of this work ................. 34
2.4 Proposition 2.7 in the easy case s = σ ................. 35
2.5 Proposition 2.7 in the hard case s < σ ................. 41
3 The Second Step: The Fefferman-Graham Ambient Metric
and the Nature of the Decomposition ......................... 71
3.1 Introduction ........................................... 71
3.2 The locally conformally invariant piece in P(g):
A proof of Lemmas 3.1, 3.2, and 3.3 .................... 78
3.3 Proof of Lemma 3.4: The divergence piece in P(g) ...... 101
4 A Result on the Structure of Local Riemannian Invariants:
The Fundamental Proposition ................................ 135
4.1 Introduction .......................................... 135
4.2 The fundamental Proposition 4.13 ...................... 136
4.3 Proof of Proposition 4.13: Set up an induction and
reduce the inductive step to Lemmas 4.16, 4.19, 4.24 .. 148
4.4 Proof that Proposition 4.13 follows from Lemmas
4.16, 4.19, and 4.24 (and Lemmas 4.22 and 4.23) ....... 162
5 The Inductive Step of the Fundamental Proposition: The
Simpler Cases .............................................. 211
5.1 Introduction .......................................... 211
5.2 Notation and preliminary results ...................... 216
5.3 An analysis of CurvTrans[Lg] .......................... 246
5.4 A study of LC[Lg] and W[Lg] in (5.16):
Computations and cancellations ........................ 273
6 The Inductive Step of the Fundamental Proposition: The
Hard Cases, Part I ......................................... 297
6.1 Introduction .......................................... 297
6.2 The first ingredient in the grand conclusion .......... 303
6.3 The second part of the grand conclusion: A study of
ImageØu+11,β[Lg] = 0 .................................... 317
6.4 The grand conclusion and the proof of Lemma 4.24 ...... 354
7 The Inductive Step of the Fundamental Proposition:
The Hard Cases, Part II .................................... 361
7.1 Introduction: A sketch of the strategy ................ 361
7.2 The proof of Lemma 4.24 in Case В ..................... 361
A Appendix ................................................... 403
A.l Some technical tools .................................. 403
A.2 Some postponed short proofs ........................... 410
A.3 Proof of Lemmas 4.22 and 4.23 ......................... 432
Bibliography .................................................. 443
Index of Authors and Terms .................................... 447
Index of Symbols .............................................. 449
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