The Ambient Metric (AM-178)

Charles Fefferman; C. Robin Graham

Charles Fefferman and C. Robin Graham

The Ambient Metric (AM-178)

In: Annals of Mathematics Studies, 178

Princeton University Press | 2012

https://doi.org/10.1515/9781400840588

Overview
Contents

About this book

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric in n+1 dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics.

The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

Author information

Charles Fefferman is the Herbert E. Jones, Jr., '43 University Professor of Mathematics at Princeton University. C. Robin Graham is professor of mathematics at the University of Washington.

Reviews

It is concise, but detailed, accurate, and comprehensive in its treatment of the topics it covers, including their technical details. The book will be of interest to anyone working in, or using, conformal geometry or closely related structures for mathematics, theoretical physics, or physical applications.---Rod Gover, SIAM Review

[T]his careful exposition has been well worth the wait!---Michael G. Eastwood, Mathematical Reviews Clippings

Nowadays, it seems that any of the more subtle advances in local conformal differential geometry depend on the ambient metric in an essential way. Such advances continue apace. The careful exposition provided by the ambient metric has been well worth the wait!---Michael Eastwood, Bulletin of the American Mathematical Society

Frontmatter Free access PDF PDF	i
Contents Free access PDF PDF	vii
Chapter One. Introduction Access restricted Content is available PDF PDF	1
Chapter Two. Ambient Metrics Access restricted Content is available PDF PDF	9
Chapter Three. Formal Theory Access restricted Content is available PDF PDF	17
Chapter Four. Poincaré Metrics Access restricted Content is available PDF PDF	42
Chapter Five. Self-dual Poincaré Metrics Access restricted Content is available PDF PDF	50
Chapter Six. Conformal Curvature Tensors Access restricted Content is available PDF PDF	56
Chapter Seven. Conformally Flat and Conformally Einstein Spaces Access restricted Content is available PDF PDF	67
Chapter Eight. Jet Isomorphism Access restricted Content is available PDF PDF	82
Chapter Nine. Scalar Invariants Access restricted Content is available PDF PDF	97
Bibliography Access restricted Content is available PDF PDF	107
Index Access restricted Content is available PDF PDF	113

(Deutsch)

The Ambient Metric (AM-178)

Details

Formats