## The Ambient Metric (AM-178) |

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in 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. "[T]his careful exposition has been well worth the wait!" "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." "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!" Chapter 1. Introduction 1
- Advances in Analysis: The Legacy of Elias M. Stein. [Hardcover]
- Essays on Fourier Analysis in Honor of Elias M. Stein (PMS-42). [Hardcover and Paperback]
- Annals of Mathematics Studies
Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Editors
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