Weyl compatible tensors

TL;DR

This paper introduces Weyl compatibility, a new algebraic property of tensors that relates to Riemann compatibility, with implications for the algebraic structure of the Weyl tensor and conditions for the magnetic part to vanish.

Contribution

It defines Weyl compatibility, explores its geometric implications, and extends existing theorems to this broader context, linking it to general relativity conditions.

Findings

Weyl compatible vectors imply algebraically special Weyl tensors.

Weyl compatibility is necessary and sufficient for the magnetic part to vanish.

Hypersurfaces of pseudo Euclidean spaces exhibit Weyl compatible Ricci tensors.

Abstract

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular it is shown that the existence of a Weyl compatible vector implies the Weyl tensor to be algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdzinski and Shen, Hall) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as McIntosh et al.). Hypersurfaces of pseudo Euclidean spaces provide a simple example of Weyl compatible Ricci tensor.