On the rigidity of generalized $m$-quasi-Einstein manifolds of Yamabe-type

Motivated by the concept of almost Yamabe solitons, a special class of generalized $m$-quasi-Einstein manifolds is investigated in this paper.

We refer to these Riemannian manifolds as generalized $m$-quasi-Einstein manifolds of Yamabe-type.

We study the rigidity properties for the potential (or defining) vector field associated to these manifolds in both the compact and non-compact settings.

We show that under certain natural assumptions the potential vector field either vanishes identically or become a non-trivial Killing vector field.