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The Ambient Metric

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Title: The Ambient Metric
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 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.
Authors: Charles Fefferman, C. Robin Graham
Resource Type: eBook.
Subjects: Conformal invariants, Conformal geometry
Categories: MATHEMATICS / Geometry / Analytic
Database: eBook Collection (EBSCOhost)
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Header DbId: nlebk
DbLabel: eBook Collection (EBSCOhost)
An: 396360
RelevancyScore: 1044
AccessLevel: 6
PubType: eBook
PubTypeId: ebook
PreciseRelevancyScore: 1044.26904296875
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: The Ambient Metric
– Name: Abstract
  Label: Description
  Group: Ab
  Data: 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.
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  Data: <searchLink fieldCode="AR" term="%22Charles+Fefferman%22">Charles Fefferman</searchLink><br /><searchLink fieldCode="AR" term="%22C%2E+Robin+Graham%22">C. Robin Graham</searchLink>
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  Data: eBook.
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  Data: <searchLink fieldCode="DE" term="%22Conformal+invariants%22">Conformal invariants</searchLink><br /><searchLink fieldCode="DE" term="%22Conformal+geometry%22">Conformal geometry</searchLink>
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  Data: <searchLink fieldCode="ZK" term="%22MATHEMATICS+%2F+Geometry+%2F+Analytic%22">MATHEMATICS / Geometry / Analytic</searchLink>
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    Classifications:
      – Code: 516.37
        Scheme: ddc
        Type: prePub
    Languages:
      – Code: eng
        Text: English
    Subjects:
      – SubjectFull: Conformal invariants
        Type: general
      – SubjectFull: Conformal geometry
        Type: general
    Titles:
      – TitleFull: The Ambient Metric
        Type: main
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      – PersonEntity:
          Name:
            NameFull: Charles Fefferman
      – PersonEntity:
          Name:
            NameFull: C. Robin Graham
      – PersonEntity:
          Name:
            NameFull: Charles Fefferman
      – PersonEntity:
          Name:
            NameFull: C. Robin Graham
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          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2012
            – D: 04
              M: 02
              Type: profile
              Y: 2014
          Identifiers:
            – Type: isbn-print
              Value: 9780691153148
            – Type: isbn-print
              Value: 9780691153131
            – Type: isbn-electronic
              Value: 9781400840588
          Numbering:
            – Type: volume
              Value: 00178
          Titles:
            – TitleFull: The Ambient Metric
              Type: main
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