On generalized quasi-Einstein manifolds

Abstract

Abstract We study generalized quasi-Einstein manifolds, or briefly, GQE manifolds. Here, we present relations between the Bach, Cotton and D tensors on GQE manifolds. Next, a 3-tensor E which measures the deviation of m-quasi-Einstein manifolds from GQE manifolds is introduced. Among others in dimension 3, it is shown that Bach-flatness implies locally conformally flatness. Furthermore, it is proved that, around a regular point of the fourth-order divergence free Weyl tensor, a GQE manifold is a locally warped product manifold with {(n-1)} -dimensional Einstein fibers in suitable cases.