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Inhomogeneous Ambient Metrics

  • C. Robin Graham
  • Kengo Hirachi
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)

Abstract

The ambient metric, introduced in [FG1], has proven to be an important object in conformal geometry. To a manifold M of dimension n with a conformai class of metrics [g] of signature (p, q) it associates a diffeomorphism class of formal expansions of metrics \( \tilde g \) of signature (p + 1, q + 1) on a space Open image in new window of dimension n + 2. This generalizes the realization of the conformal sphere Sn as the space of null lines for a quadratic form of signature (n + 1, 1), with associated Minkowski metric \( \tilde g \) on ℝn+2. The ambient space Open image in new window carries a family of dilations with respect to which \( \tilde g \) is homogeneous of degree 2. The other conditions determining \( \tilde g \) are that it be Ricci-flat and satisfy an initial condition specified by the conformal class [g].

Keywords

Normal Form Bergman Kernel Conformal Class Smooth Part Complete Contraction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • C. Robin Graham
    • 1
  • Kengo Hirachi
    • 2
  1. 1.Department of MathematicsUniversity of WashingtonSeattle
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

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