On the h-almost Ricci soliton

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.