Extended Derdzinski-Shen theorem for the Riemann tensor

TL;DR

This paper extends a classical theorem relating to the Riemann tensor by incorporating broader classes of Codazzi tensors, including recurrent forms, thus generalizing the restrictions on the Riemann tensor.

Contribution

It generalizes the Derdzinski-Shen theorem to include tensors with gauged Codazzi conditions, broadening its applicability to various differential structures.

Findings

Extended the theorem to include recurrent 1-forms

Provided new conditions for the Riemann tensor restrictions

Broadened the class of Codazzi tensors considered

Abstract

We extend a classical result by Derdzinski and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms) as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"), typical of some well known differential structures.