On Generalized Quasi-Einstein Manifolds

TL;DR

This paper investigates generalized m-quasi-Einstein manifolds, proving potential vector fields are Killing under certain conditions, and explores related rigidity and divergence-free vector field results.

Contribution

It extends previous results by showing potential vector fields are Killing under integral conditions and revisits key theorems with new proofs.

Findings

Potential vector fields are Killing under integral assumptions.

Divergence-free vector fields are Killing in this setting.

Rigidity results for manifolds with geodesic potential vector fields.

Abstract

In this paper, we study generalized $m$-quasi-Einstein $(M^n,g,X,\lambda)$ under natural conditions on the potential vector field. We show that, under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on $m$ and $\lambda$, including triviality results. We also revisit a recent theorem of Ghosh \cite{ghosh}, discuss a subtle issue in the argument, and provide a new formulation and proof. Finally, we establish rigidity results for manifolds with geodesic potential vector fields.