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 Poincare 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. Poincare metrics are introduced and shown to be
equivalent to the ambient formulation. Self-dual Poincare 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."
General
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