Rigidity of (m,ρ)‐quasi Einstein manifolds

This paper deals with the study on (m,ρ)‐quasi Einstein manifolds. First, we give some characterizations of an (m,ρ)‐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,ρ)‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form I×eq/2M∗, where I is a real interval, (M∗,g∗) is an (n−1)‐dimensional Riemannian manifold and q is a smooth function on I. Finally, a non‐trivial example of an (m,ρ)‐quasi Einstein manifold verifying our results in terms of the potential function is presented.