Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

TL;DR

This paper investigates the behavior of quasi-Einstein metrics under the Ricci flow, establishing rigidity results for certain closed quasi-Einstein manifolds using curvature evolution identities.

Contribution

It introduces a curvature identity related to Ricci flow evolution and applies it to prove rigidity of specific closed quasi-Einstein manifolds.

Findings

Established a new curvature identity for Ricci flow.

Proved rigidity results for certain closed quasi-Einstein manifolds.

Abstract

Quasi-Einstein manifolds are well-studied generalizations of Einstein manifolds. This includes gradient Ricci solitons and has a natural correspondence with the warped product Einstein manifolds. A quasi-Einstein metric is said to be rigid when it reduces to an Einstein metric. On a different note, Einstein metrics can be viewed as fixed points of the Ricci flow up to homothety. While gradient Ricci solitons are generalized fixed points of the Ricci flow, not much is known, in general, about the evolution of quasi-Einstein metrics under the Ricci flow. In this paper, we employ an identity associated to the evolution of curvature along the Ricci flow, to conclude the rigidity of certain closed quasi-Einstein manifolds.