Rigidity of almost Ricci solitons on compact Riemannian manifolds

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.