Almost Ricci solitons on Finsler spaces

Abstract: 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 if and only if the infinity-Ricci curvature Ric$\infty$ is a scalar function on $M$ when $M$ is compact. Moreover, we give an equivalent characterization of (gradient) almost Ricci solitons for Randers metrics $F=\alpha+\beta$, 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.