On a class of Finsler gradient Ricci solitons

Abstract

In this paper, we study a class of Finsler measure spaces whose weighted Ricci curvature satisfies R i c ∞ = c F 2 {\mathbf {Ric}}_{\infty }=cF^{2} . This class contains all gradient Ricci solitons and Finsler Gaussian shrinking solitons. Thus Finsler measure spaces in this class are called Finsler gradient Ricci solitons. For a Randers measure space, we find sufficient and necessary conditions for this space to be a Finsler gradient Ricci soliton. In particular, we show that Randers-Finsler gradient Ricci solitons must have isotropic S S -curvature. Finally, we give an equivalent condition for a Randers measure space to be a Finsler gradient Ricci soliton of constant S S -curvature.