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.
