The $\mathcal{C}$-connection and the 4-dimensional Einstein spaces

TL;DR

This paper introduces the $ abla$-connection in 4D pseudo-Riemannian manifolds, providing a new way to characterize conformally Einstein spaces through conformally covariant tensor operators.

Contribution

It defines the $ abla$-connection that preserves conformal covariance and offers a novel characterization of non-degenerate spaces conformal to Einstein spaces.

Findings

The $ abla$-connection maintains conformal invariance of tensor concomitants.

A new criterion for identifying conformally Einstein spaces is established.

The approach applies to 4D pseudo-Riemannian manifolds with non-zero Weyl scalar.

Abstract

We introduce a new family of operators in 4-dimensional pseudo-Riemannian manifolds with a non-vanishing Weyl scalar (non-degenerate spaces) that keep the conformal covariance of \emph{conformally covariant tensor concomitants}. A particular case that arises naturally is the $\mathcal{C}$-connection that is a Weyl connection that keeps \emph{conformal invariance}. Using the $\mathcal{C}-$connection we give a new characterization of non-degenerate spaces that are conformal to an Einstein space.