A splitting theorem for Riemannian manifolds of generalised Ricci-Hessian type

TL;DR

This paper investigates Riemannian manifolds with a special Hessian-Ricci tensor relation, providing partial classification results for manifolds with a non-trivial function satisfying this geometric condition.

Contribution

It introduces a splitting theorem for Riemannian manifolds of generalized Ricci-Hessian type, advancing understanding of their geometric structure.

Findings

Partial classification of manifolds with the specified Hessian-Ricci relation

Identification of conditions under which the manifold splits

New geometric insights into Ricci-Hessian type manifolds

Abstract

In this paper, we study and partially classify those Riemannian man-ifolds carrying a non-identically vanishing function f whose Hessian is minus f times the Ricci-tensor of the manifold.