The ambient metric / Charles Fefferman, C. Robin Graham.

Format:

Book

Author/Creator:

Fefferman, Charles, 1949-

Contributor:

Graham, C. Robin, 1954-

Series:

Annals of mathematics studies ; no. 178.

Annals of mathematics studies ; no. 178

Language:

English

Subjects (All):

Conformal geometry.

Conformal invariants.

Physical Description:

1 online resource (124 p.)

Edition:

Course Book

Place of Publication:

Princeton : Princeton University Press, 2012.

Language Note:

English

Summary:

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.

Contents:

Front matter

Contents

Chapter One. Introduction

Chapter Two. Ambient Metrics

Chapter Three. Formal Theory

Chapter Four. Poincaré Metrics

Chapter Five. Self-dual Poincaré Metrics

Chapter Six. Conformal Curvature Tensors

Chapter Seven. Conformally Flat and Conformally Einstein Spaces

Chapter Eight. Jet Isomorphism

Chapter Nine. Scalar Invariants

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786613290953

9781283290951

1283290952

9781400840588

1400840589

OCLC:

768165026