M-quasi-Einstein metric and contact geometry

Abstract

We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of $$ \eta $$ -Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field $$\xi $$ . Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided $$m\ne 1$$ . We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is $$ \eta $$ -Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.