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## Spyros Alexakis

Print publication date: 2012

Print ISBN-13: 9780691153476

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153476.001.0001

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# The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II

Chapter:
(p.361) Chapter Seven The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II
Source:
The Decomposition of Global Conformal Invariants (AM-182)
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153476.003.0007

This chapter takes up the proof of Lemma 4.24 in Case B; this completes the inductive step of the proof of the fundamental proposition 4.13. The chapter recalls that Lemma 4.24 applies when all tensor fields of minimum rank μ‎ in (4.3) have all μ‎ of their free indices being nonspecial. We recall the setting of Case B in Lemma 4.24: Let M > 0 stand for the maximum number of free indices that can belong to the same factor, among all tensor fields in (4.3). Then consider all μ‎-tensor fields in (4.3) that have at least one factor T₁ containing M free indices; let M' ≤ M be the maximum number of free indices that can belong to the same factor, other than T₁. The setting of Case B in Lemma 4.24 is when M < 2. The chapter recalls that in the setting of case B the claim of Lemma 4.24 coincides with the claim of Proposition 4.13. To derive Lemma 4.24, the authors use all the tools developed in the Chapter 6, most importantly the grand conclusion, but also the two separate equations that were added to derive the grand conclusion.

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