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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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Introduction

Introduction

Chapter:
(p.1) Chapter One Introduction
Source:
The Ambient Metric (AM-178)
Author(s):

Charles Fefferman

C. Robin Graham

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153131.003.0001

This introductory chapter begins with a brief definition of conformal geometry. Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. In [FG], the authors outlined a construction of a nondegenerate Lorentz metric in n+2 dimensions associated to an n-dimensional conformal manifold, which they called the ambient metric. This association enables one to construct conformal invariants in n dimensions from pseudo-Riemannian invariants in n+2 dimensions, and in particular shows that conformal invariants are plentiful. The formal theory outlined in [FG] did not provide details. This book provides these details. An overview of the subsequent chapters is also presented.

Keywords:   conformal geometry, ambient metric, Riemannian metrics, manifold, Lorentz metric, conformal invariants

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