Almost Ricci Solitons on Finsler Spaces

In this paper, (gradient) almost Ricci solitons on Finsler measure spaces ( M , F , m ) are introduced and investigated. We prove that ( M , F , m ) is a gradient almost Ricci soliton with soliton scalar κ if and only if the infinity-Ricci curvature Ric ∞ = κ on M . Moreover, we give an equivalent characterization of (gradient) almost Ricci solitons for Randers metrics F = α + β , which implies that every Randers (gradient) almost Ricci soliton is of isotropic S BH -curvature. Based on this and the navigation technique, we further classify Randers almost Ricci solitons (resp., gradient almost Ricci solitons) up to classifications of Randers Einstein metrics F (resp., Riemannian gradient almost Ricci solitons) and the homothetic vector fields of F (resp., solutions of the equation which the weight function f of m satisfies) when F has isotropic S BH -curvature. As applications, we obtain some rigidity results for compact Randers (gradient) Ricci solitons and construct several Randers gradient Ricci solitons, which are the first nontrivial examples of gradient Ricci solitons in Finsler geometry.