On Riemannian and Ricci curvatures of homogeneous Finsler manifolds

Abstract The famous Cheng-Shen’s conjecture in Riemann-Finsler geometry claims that every n -dimensional closed W-quadratic Randers manifold is a Berwald manifold. In this paper, first we study the Riemann and Ricci curvatures of homogeneous Finsler manifolds and obtain some rigidity theorems. Then, by using this investigation, we construct a family of W-quadratic Randers metrics which are not R-quadratic nor strongly Ricci-quadratic.