Rigidity of gradient almost Ricci solitons

Abstract

In this paper, we show that either, a Euclidean space $\mathbb{R}^{n}$, or a standard sphere $\mathbb{S}^{n}$, is the unique manifold with nonnegative scalar curvature which carries a structure of a gradient almost Ricci soliton, provided this gradient is a non trivial conformal vector field. Moreover, in the spherical case the field is given by the first eigenfunction of the Laplacian. Finally, we shall show that a compact locally conformally flat almost Ricci soliton is isometric to Euclidean sphere $\mathbb{S}^{n}$ provided an integral condition holds.