Compact almost Ricci solitons with constant scalar curvature are gradient

Abstract

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $(M^n,g,X,\lambda )$ with constant scalar curvature is isometric to a Euclidean sphere ${\mathbb{S}}^n$. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.

To appear in Monatshefte fur Mathematik