Compact almost Ricci solitons with constant scalar curvature are gradient

TL;DR

This paper proves that compact, non-trivial almost Ricci solitons with constant scalar curvature are isometric to spheres and are necessarily gradient solitons, with the vector field decomposing into a Killing field and a gradient.

Contribution

It establishes that such solitons are isometric to spheres and confirms their gradient nature, extending understanding of Ricci solitons with constant scalar curvature.

Findings

Compact almost Ricci solitons with constant scalar curvature are spherical.

These solitons are necessarily gradient Ricci solitons.

The vector field decomposes into a Killing vector and a 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 $\Bbb{S}^{n}$. 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.