Generalized m-quasi-Einstein metrics with certain conditions on the potential vector field

In this paper, we have studied generalized m-quasi-Einstein and m-quasi-Einstein manifold ($M^n,g,X,\lambda )$ satisfying certain conditions on the potential vector field. First, we classify a compact m-quasi-Einstein metric with geodesic potential and non-positive Ricci tensor. Then, we establish that the potential vector field X of an m-quasi-Einstein metric is Killing if X is an infinitesimal harmonic transformation and geodesic. Further, we classify complete non-compact m-quasi-Einstein metric with Ricci tensor $S(X,X)\le 0$ when the potential vector field X is an infinitesimal harmonic transformation, or its energy is finite. Furthermore, we establish that the potential vector field X of a compact generalized m-quasi-Einstein manifold is Killing when it satisfies $divX=0$. Finally, we prove that a generalized m-quasi-Einstein manifold is a warped product when its potential vector field is a non-homothetic conformal Killing.