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The ambient metric

Charles Fefferman, C. Robin Graham

TL;DR

The ambient metric framework of Fefferman and Graham provides a canonical higher-dimensional setting in which conformal invariants of a manifold can be captured as pseudo-Riemannian invariants. The authors develop a comprehensive formal theory for ambient metrics, including existence, straightness, normal form, and the parity-dependent obstruction tensor, and connect ambient metrics to Poincaré metrics. They introduce conformal curvature tensors, establish a jet isomorphism linking conformal jets to ambient-curvature data, and deploy invariant theory to characterize scalar conformal invariants as Weyl invariants in most cases, with notable exceptions tied to Pontrjagin-type invariants in certain even dimensions. The work further analyzes special cases (locally conformally flat and conformally Einstein) and derives concrete consequences for GJMS operators and Q-curvature, underpinning both geometric analysis and AdS/CFT-type physics applications. Overall, the paper provides a unifying, algebraic-analytic approach to conformal invariants via higher-dimensional ambient geometry and parabolic invariant theory.

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.

The ambient metric

TL;DR

The ambient metric framework of Fefferman and Graham provides a canonical higher-dimensional setting in which conformal invariants of a manifold can be captured as pseudo-Riemannian invariants. The authors develop a comprehensive formal theory for ambient metrics, including existence, straightness, normal form, and the parity-dependent obstruction tensor, and connect ambient metrics to Poincaré metrics. They introduce conformal curvature tensors, establish a jet isomorphism linking conformal jets to ambient-curvature data, and deploy invariant theory to characterize scalar conformal invariants as Weyl invariants in most cases, with notable exceptions tied to Pontrjagin-type invariants in certain even dimensions. The work further analyzes special cases (locally conformally flat and conformally Einstein) and derives concrete consequences for GJMS operators and Q-curvature, underpinning both geometric analysis and AdS/CFT-type physics applications. Overall, the paper provides a unifying, algebraic-analytic approach to conformal invariants via higher-dimensional ambient geometry and parabolic invariant theory.

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.

Paper Structure

This paper contains 9 sections, 60 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Ambient Metrics
  3. Formal theory
  4. Poincaré Metrics
  5. Self-dual Poincaré Metrics
  6. Conformal Curvature Tensors
  7. Conformally Flat and Conformally Einstein Spaces
  8. Jet Isomorphism
  9. Scalar Invariants

Key Result

Theorem 2.3

Let $(M, [ g ])$ be a smooth manifold of dimension $n\geq 2$, equipped with a conformal class. Then there exists an ambient space for $(M , [g])$. Moreover, any two ambient spaces for $(M , [g])$ are ambient-equivalent.

Theorems & Definitions (109)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • Lemma 2.10
  • ...and 99 more