Some Properties of the Potential Field of an Almost Ricci Soliton

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c , the integral of Ric ( c ξ , c ξ ) satisfying a generic inequality on an n -dimensional compact and connected almost Ricci soliton ( M n , g , ξ , σ ) are necessary and sufficient conditions for it to be isometric to the n -sphere S n ( c ) . As another result, we show that, if the affinity tensor of the soliton vector field ξ vanishes and the scalar curvature τ of an n -dimensional compact almost Ricci soliton ( M n , g , ξ , σ ) satisfies τ n σ − τ ≥ 0 , then ( M n , g , ξ , σ ) is a trivial Ricci soliton. Finally, on an n -dimensional compact almost Ricci soliton ( M n , g , ξ , σ ) , we consider the Hodge decomposition ξ = ξ ¯ + ∇ h , where div ξ ¯ = 0 , and we use the bound on the integral of Ric ξ ¯ , ξ ¯ and an integral inequality involving the scalar curvature to find another characterization of the n -sphere.