Obstructions to conformally Einstein metrics in $n$ dimensions

TL;DR

This paper develops polynomial conformal invariants that precisely characterize when a (pseudo-)Riemannian manifold is conformally Einstein, extending previous invariants and linking them to tractor calculus and ambient metrics.

Contribution

It introduces new polynomial invariants for conformally Einstein metrics, generalizes existing invariants, and connects them to tractor calculus and ambient metric curvature.

Findings

Constructed necessary and sufficient invariants for conformally Einstein metrics.

Extended invariants to a broader class of metrics beyond previous results.

Linked invariants to tractor connection and ambient metric curvature.

Abstract

We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an $n$-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally related to a metric with vanishing Cotton tensor. One set of invariants we derive generalises the set of invariants in dimension four obtained by Kozameh, Newman and Tod. For the conformally Einstein problem, another set of invariants we construct gives necessary and sufficient conditions for a wider class of metrics than covered by the invariants recently presented by M. Listing. We also show that there is an alternative characterisation of conformally Einstein metrics based on the tractor connection associated with the normal conformal Cartan bundle. This plays a key role in constructing some of the invariants. Also using this we can interpret the previously known invariants geometrically in the tractor setting and relate some of them to the curvature of the Fefferman-Graham ambient metric.