Riemann compatible tensors

TL;DR

This paper generalizes the concept of Codazzi tensors to Riemann-compatible tensors, exploring their properties, geometric significance, and applications in manifolds and general relativity.

Contribution

It extends Derdzinski and Shen's theorem to Riemann-compatible tensors, establishing their properties and relevance to curvature tensors and Einstein's theory.

Findings

Riemann compatibility relates to the Bianchi identity of the Codazzi deviation tensor

Examples include manifolds generated by geodesic mapping

Compatibility extends to generalized curvature tensors like Weyl's tensor

Abstract

Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation tensor" with a geometric significance. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mapping. Compatibility is extended to generalized curvature tensors with an application to Weyl's tensor and general relativity.