
The Ambient Metric
Princeton University Press
Published on 4. December 2011
Book
Paperback/Softback
128 pages
978-0-691-15314-8 (ISBN)
Description
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 Poincar 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. Poincar metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincar 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.
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.
Reviews / Votes
"[T]his careful exposition has been well worth the wait!"--Michael G. Eastwood, Mathematical Reviews Clippings "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 "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 SocietyMore details
Series
Language
English
Place of publication
New Jersey
United States
Target group
College/higher education
Professional and scholarly
Product notice
Paperback (trade)
Dimensions
Height: 234 mm
Width: 156 mm
Thickness: 8 mm
Weight
202 gr
ISBN-13
978-0-691-15314-8 (9780691153148)
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Schweitzer Classification
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Charles Fefferman | C. Robin Graham
The Ambient Metric
Book
12/2011
Princeton University Press
€86.66
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Charles Fefferman | C. Robin Graham
The Ambient Metric
E-Book
11/2011
1st Edition
Princeton University Press
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Persons
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.
Content
Chapter 1. Introduction 1 Chapter 2. Ambient Metrics 9 Chapter 3. Formal Theory 17 Chapter 4. Poincar'e Metrics 42 Chapter 5. Self-dual Poincar'e Metrics 50 Chapter 6. Conformal Curvature Tensors 56 Chapter 7. Conformally Flat and Conformally Einstein Spaces 67 Chapter 8. Jet Isomorphism 82 Chapter 9. Scalar Invariants 97 Bibliography 107 Index 113