On Generalized Quasi-Einstein Manifolds

Abstract

In this paper, we study generalized $m$-quasi-Einstein $(M^n,g,X,\hat{I}»)$ 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 $\hat{I}»$, 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.

The revised version improves the manuscript by clarifying the main rigidity results, correcting the discussion surrounding Ghosh's theorem, and providing a counterexample showing that a previous generalized statement is false in full generality. We are grateful to Ghosh for clarify several points in his recent paper, which improve the presentation and a precise formulation our results