Standard Tractors and the Conformal Ambient Metric Construction

TL;DR

This paper connects the ambient metric construction in conformal geometry with the canonical Cartan connection, providing new methods to derive conformal invariants and insights into Ricci-flat ambient metrics.

Contribution

It establishes a link between ambient metrics and the Cartan approach, enabling the construction of tractor bundles and formulae for conformal invariants.

Findings

Constructed conformal standard tractor bundle from ambient metrics.

Derived tractor formulae for conformal invariants.

Provided new insights into higher-order Ricci-flat ambient metrics.

Abstract

In this paper we relate the Fefferman-Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to the Cartan bundle and the Cartan connection. This result is applied to obtain a procedure to get tractor formulae for all conformal invariants that can be obtained from the ambient metric construction. We also get information on ambient metrics which are Ricci flat to higher order than guaranteed by the results of Fefferman-Graham.