Generalized m-Quasi-Einstein manifolds admitting a closed conformal vector field

Abstract

We study a complete connected generalized m-quasi-Einstein manifold M with finite m, admitting a non-homothetic, non-parallel, complete closed conformal vector field V, and show that either M is isometric to a round sphere, or the Ricci tensor can be expressed explicitly in terms of the conformal data over an open dense subset. In the latter case, we prove that M is a warped product of an open real interval with an Einstein manifold; furthermore, it is conformally flat in dimension 4 and has vanishing Cotton and Bach tensors in dimension > 3. Next, we obtain the same explicit expression for the Ricci tensor, and analogous results, for a gradient Ricci almost soliton endowed with a non-parallel closed conformal vector field.