On the geometry of gradient Einstein-type manifolds

In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\text{varrg}$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.