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The Ambient Metric (AM-178)$
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Charles Fefferman and C. Robin Graham

Print publication date: 2011

Print ISBN-13: 9780691153131

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153131.001.0001

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Jet Isomorphism

Jet Isomorphism

Chapter:
(p.82) Chapter Eight Jet Isomorphism
Source:
The Ambient Metric (AM-178)
Author(s):

Charles Fefferman

C. Robin Graham

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153131.003.0008

A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors. This chapter proves an analogous jet isomorphism theorem for conformal geometry. By making conformal changes, the Taylor expansion of a metric in geodesic normal coordinates can be further simplified, resulting in a “conformal normal form” for metrics about a point.

Keywords:   jet isomorphism theorem, conformal geometry, ambient metric, Riemannian geometry, geodesic normal coordinates, Taylor expansion

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