The ambient metric

TL;DR

This paper explores the construction, properties, and applications of the ambient metric in conformal geometry, including existence, uniqueness, and its relation to Poincare metrics, along with conformal curvature tensors and invariants.

Contribution

It introduces a detailed construction of ambient metrics, establishes their equivalence with Poincare metric expansions, and characterizes conformal invariants using jet isomorphism theorems.

Findings

Existence and uniqueness of formal ambient metric expansions

Equivalence between ambient and Poincare metric expansions

Characterization of scalar conformal invariants

Abstract

This paper provides details of the construction, properties and some applications of the ambient metric associated to a conformal class of metrics on a smooth manifold. Existence and uniqueness of formal expansions defining such metrics are considered. Equivalence with the expansions of associated Poincare metrics is established. Definitions and properties of conformal curvature tensors defined by ambient metrics together with formulation and proof of a jet isomorphism theorem with application to the characterization of scalar conformal invariants are given.