A note on the coincidence of the projective and conformal Weyl tensors

TL;DR

This paper investigates when the projective and conformal Weyl tensors are equal, showing that for dimensions greater than three, this occurs if and only if the connection is the Levi-Civita connection of an Einstein metric.

Contribution

It establishes a precise condition linking the coincidence of Weyl tensors to Einstein metrics in higher dimensions.

Findings

Weyl tensors coincide iff D is Levi-Civita of an Einstein metric for n>3.

The result applies to general Weyl connections associated with conformal classes.

Provides a characterization of Einstein metrics via Weyl tensor coincidence.

Abstract

This article examines the coincidence of the projective and conformal Weyl tensors associated to a given connection D. The connection may be a general Weyl connection associated to a conformal class of metrics [g]. The main result for n>3 is that the Weyl tensors coincide iff D is the Levi-Civita connection of an Einstein metric.