This monograph is devoted to the study of invariants of conformal classes of (semi-riemannian) metrics in differentiable manifolds. The title, "The Ambient Metric", refers to the object studied in the book. The ambient metric associated to a conformal structure on a manifold was originally introduced by the authors in a short article without complete proofs in 1985, see [C. Fefferman and C.R. Graham, Conformal invariants, in The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque, 1985, Numero Hors Serie, 95-116].
The book develops and applies the theory of the ambient metric in conformal geometry. The ambient metric is a (semi-riemannian) metric in $n+2$ dimensions that encodes the information of a conformal class of metrics in $n$ dimensions. The ambient metric has an alternative incarnation as the Poincaré metric, which is a metric in $n+1$ dimensions having the conformal manifold as its conformal infinity.
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. Then Poincaré metrics are introduced and shown to be equivalent to the ambient metric formulation. The special case of self-dual Poincaré metrics in four dimensions is considered, leading to a formal power series proof of LeBrun's collar neighborhood theorem, proved originally using twistor methods, for analytic metrics.
Conformal curvature tensors are introduced by using the ambient metric formulation, and their fundamental properties are established. The book concludes with the construction and characterization of scalar conformal invariants in terms of the curvature of the ambient metric.
The monograph is a text of technical nature, very carefully written, which includes full proofs. it is focused on a particular problem in geometry. As such, it is addressed to researchers already interested in this area of mathematics.