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(cξ,cξ) satisfying a generic inequality on an n-dimensional compact and connected almost Ricci soliton (Mn,g,ξ,σ) are necessary and sufficient conditions for it to be isometric to the n-sphere Sn(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 (Mn,g,ξ,σ) satisfies τnσ−τ≥0, then (Mn,g,ξ,σ) is a trivial Ricci soliton. Finally, on an n-dimensional compact almost Ricci soliton (Mn,g,ξ,σ), we consider the Hodge decomposition ξ=ξ¯+∇h, where divξ¯=0, and we use the bound on the integral of Ricξ¯,ξ¯ and an integral inequality involving the scalar curvature to find another characterization of the n-sphere.