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The Decomposition of Global Conformal Invariants (AM-182)$
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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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An Iterative Decomposition of Global Conformal Invariants: The First Step

An Iterative Decomposition of Global Conformal Invariants: The First Step

Chapter:
(p.19) Chapter Two An Iterative Decomposition of Global Conformal Invariants: The First Step
Source:
The Decomposition of Global Conformal Invariants (AM-182)
Author(s):

Spyros Alexakis

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153476.003.0002

This chapter fleshes out the strategy of iteratively decomposing any P(g) = unconverted formula 1 for which ∫P(g)dVsubscript g is a global conformal invariant. It makes precise the notions of better and worse complete contractions in P(g) and then spells out (1.17), via Propositions 2.7, 2.8. In particular, using the well-known decomposition of the curvature tensor into its trace-free part (the Weyl tensor) and its trace part (the Schouten tensor), it reexpresses P(g) as a linear combination of complete contractions involving differentiated Weyl tensors and differentiated Schouten tensors, as in (2.47). The chapter also proves (1.17) when the worst terms involve at least one differentiated Schouten tensor.

Keywords:   iterative decomposition, global conformal invariant, curvature tensor, Weyl tensor, Schouten tensor

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