A note on rigidity of the almost Ricci soliton

摘要

The aim of this paper is to prove that a gradient almost Ricci soliton (M~n, g, ?f, Λ) whose Ricci tensor is Codazzi has constant sectional curvature. In particular, in the compact case, we deduce that (M~n, g) is isometric to a Euclidean sphere and f is a height function. Moreover, we also classify gradient almost Ricci solitons with constant scalar curvature R provided a suitable function achieves a maximum in M~n.