On the h-almost Ricci soliton

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We introduce the concept h-almost Ricci soliton which extends naturally the almost Ricci soliton by Pigola–Rigoli–Rimoldi–Setti and show that a compact nontrivial h -almost Ricci soliton of dimension no less than three with h having defined signal and constant scalar curvature is isometric to a standard sphere with the potential function well determined. We also consider the h-Ricci soliton which is a particular case of the h -almost Ricci soliton and a generalization of the Ricci soliton and give characterizations for a special class of gradient h -Ricci solitons.