Conformal Einstein spaces and conformally covariant operators

TL;DR

This paper provides criteria for when a pseudo-Riemannian manifold is conformally Einstein, based on the metric and Weyl endomorphism, and introduces conformally covariant operators with potential applications.

Contribution

It establishes necessary and sufficient conditions for conformal Einstein spaces and constructs new conformally covariant pseudo-differential operators.

Findings

Criteria depend on the invertibility of the Weyl endomorphism.

Conditions are algorithmic in the metric tensor.

Introduces conformally covariant pseudo-differential operators.

Abstract

In this article we give general neccessary and sufficient conditions to ensure that a pseudo-Riemannian manifold is conformal to an Einstein space. These conditions are algorithmic in \emph{the metric tensor} whenever the Weyl endomorphism is invertible. Our conditions depend in an essential manner on the $\mathcal{C}$-connection. We also show how to construct \emph{conformally covariant, pseudo-differential} operators which has an independent interest.