Generalized $m$-quasi-Einstein metric on certain almost contact manifolds

Abstract

In this paper, we study generalized $m$-quasi-Einstein metric in the context of contact geometry. First, we prove if an $H$-contact manifold admits a generalized $m$-quasi-Einstein metric with non-zero potential vector field $V$ collinear with $\xi$, then $M$ is $K$-contact and $\eta$-Einstein. Moreover, it is also true when $H$-contactness is replaced by completeness under certain conditions. Next, we prove that if a complete $K$-contact manifold admits closed generalized $m$-quasi-Einstein metric whose potential vector field is contact then $M$ is compact, Einstein and Sasakian. Finally, we obtain some results on 3-dimensional normal almost contact manifold admitting generalized $m$-quasi-Einstein metric.