In this paper, we will give a complete classification of homogeneous Randers spaces with isotropic S-curvature and positive flag curvature. This results in a large class of Finsler spaces with non-constant positive flag curvature. At the final part of the paper, we prove a rigidity result asserting that a homogeneous Randers space with almost isotropic S-curvature and negative Ricci scalar must be Riemannian. Mathematics Subject Classification (2000)53C60 · 58B20 · 22E46 IntroductionThe purpose of this paper is to give a classification of homogeneous Randers spaces with (almost) isotropic S-curvature and positive flag curvature. Our motivation to study this problem comes from the same problem in Riemannian geometry. One of the central problem in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional-, Ricci-or scalar curvature are. Up to now this has been fairly successful in scalar curvature. Also, a large numbers of interesting examples of Riemannian manifolds with positive Ricci curvature have been constructed. So far, the only obstruction to have Supported by NSFC (No.

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In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces G/H admitting G-invariant Finsler metrics with positive flag curvature. It turns out that the classification list coincides with that of the even dimensional homogeneous Riemannian manifolds with positive sectional curvature obtained by N.R. Wallach. We also find out all the coset spaces admitting invariant non-Riemannian Finsler metrics with positive flag curvature.

Abstract. In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

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