Rigidity of almost Ricci solitons on compact Riemannian manifolds

Abstract

Considering an almost Ricci soliton (ARS) $\left(N, g, \eta, \kappa \right)$ on a compact Riemannian manifold $(N, g)$, we use the Ricci curvature in the direction of the potential vector field $\eta$ to derive necessary and sufficient conditions for $(N, g)$ to be isometric to a sphere. This expands on several recent results regarding Ricci solitons and almost Ricci solitons by applying specific integral inequalities involving the Ricci curvature evaluated in the direction $\eta$. Furthermore, we present conditions under which $\eta$ is either Killing or parallel; in particular, the ARS is trivial