The ambient metric (0710.0919v2)

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 paper introduces an innovative ambient metric construction to systematically derive conformal invariants from n+2 dimensional pseudo-Riemannian invariants.
• It details the method for deriving Poincaré metrics and explains the link between ambient metrics and conformal curvature tensors.
• The jet isomorphism theorem demonstrates that scalar conformal invariants can be expressed via conformal curvature tensors, offering insights into conformally Einstein and flat manifolds.

Analysis of "The Ambient Metric"

The paper "The Ambient Metric" by Charles Fefferman and C. Robin Graham presents a comprehensive examination of ambient metrics and their significance in conformal geometry. The concept of conformal invariance forms the backbone of this exploration, with the authors proposing new techniques to construct conformal invariants derived from pseudo-Riemannian invariants in higher dimensions. Specifically, this paper considers a nondegenerate Lorentz metric in $n+2$ dimensions associated with an $n$-dimensional conformal manifold, which the authors term the "ambient metric."

The initial sections introduce the notion of the ambient metric and its utility in generating conformal invariants, particularly scalar invariants. The authors argue that conformal invariants are plentiful, establishing a framework to demonstrate this through the ambient metric, which also allows for the derivation of new conformal invariants.

Framework and Key Contributions:

1. Ambient Metric Construction:
   • The ambient metric is associated with an $n$-dimensional conformal manifold and is used to construct conformal invariants from pseudo-Riemannian invariants in two higher dimensions. This construction is pivotal for extending the understanding of conformal invariants.
2. Detailed Exposition on Poincaré Metrics:
   • The paper outlines a method to derive Poincaré metrics, which are naturally related to the ambient metrics, offering further insights into their geometric implications. This chapter establishes the "projectively compact" and "conformally compact" formulations of Poincaré metrics that bridge the relationship to hyperbolic spaces.
3. Conformal Invariance of Curvature:
   • Key relations and identities between covariant derivatives of curvature tensors are derived, which are tied to the notion of conformal curvature tensors. These tensors display solely the dependency on conformal classes rather than specific metrics, leading to extending results about conformally Einstein metrics and conformally flat manifolds.
4. Jet Isomorphism Theorem:
   • A standout result of this work is the jet isomorphism theorem. This theorem confirms that a scalar conformal invariant can be expressed in terms of conformal curvature tensors as derived from jets of metrics in a conformal normal form. The theorem is framed under the Jazz-style interpretation where the structure of a jet space in terms of differential operations aligns with curvature tensor data.
5. Resolving Long-Standing Invariances:
   • The paper speculates and proves that for odd dimensions, every scalar conformal invariant arises as a Weyl invariant constructed from the ambient metric. Additionally, it classifies exceptional invariants for even dimensions as those which do not conform to this structure.
6. Conclusion and Future Work:
   • The work affirms that an invariant theory for conformal structures in odd dimensions aligns well with classical invariant theory. Future adaptations could extend these identities and frameworks to more general pseudo-Riemannian contexts or to different geometric structures (e.g., CR geometry), potentially invoking deeper ties to the AdS/CFT correspondence in physics.

This paper elucidates a refined construction of ambient metrics and their implications in understanding the geometry of conformal manifolds, thus advancing foundational methods for constructing conformal invariants. The theoretical apparatus developed here holds substantial promise for interpreting sophisticated geometries in physics and differential geometry, with potential to drive future inquiries into higher-order invariant constructions and their applications in diverse mathematical disciplines.