Generalized quasi-Einstein manifolds with harmonic Weyl tensor

In this paper we introduce the notion of generalized quasi–Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi–Einstein manifolds. We prove that a complete generalized quasi–Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a warped product with $(n-1)$–dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.