Rigidity of (m,ρ)-quasi Einstein manifolds

Abstract

Summary: This paper deals with the study on \((m,\rho)\)-quasi Einstein manifolds. First, we give some characterizations of an \((m,\rho)\)-quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an \((m,\rho)\)-quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form \(I\times_{e^{q/2}}M^\ast\), where \(I\) is a real interval, \((M^\ast,g^\ast)\) is an \((n-1)\)-dimensional Riemannian manifold and \(q\) is a smooth function on \(I\). Finally, a non-trivial example of an \((m,\rho)\)-quasi Einstein manifold verifying our results in terms of the potential function is presented.