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 $$\big (M^n,\,g,\,X,\,\lambda \big )$$ 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.