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The aim of this paper is to study new classes of Riemannian manifolds endowedwith a smooth potential function, including in a general framework classicalcanonical structures such as Einstein, harmonic curvature and Yamabe metrics,and, above all, gradient Ricci solitons. For the most rigid cases we give acomplete classification, while for the others we provide rigidity andobstruction results, characterizations and nontrivial examples. In the finalpart of the paper we also describe the "nongradient" version of thisconstruction.

A potential generalization of some canonical Riemannian metrics