Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature

Abstract

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.

On the flag curvature of a class of Finsler metrics produced by the navigation problem

“…x p/. Recently, one of the important approaches in discussing Finsler metrics is the (Zermelo) navigation problem Hu and Deng 2012;Huang and Mo 2011;Zermelo 1931;Xia 2013]. The main technique of the navigation problem is described as follows.…”

“…In particular, the navigation problem has the flag curvature preserving property for a Killing field. Applying this result, Hu and Deng [2012] established a principle to classify homogeneous Randers spaces with (almost) isotropic S-curvature and positive flag curvature, and then they gave a complete classification of these homogeneous Randers spaces.…”

Left invariant Randers metrics on the 3-dimensional Heisenberg group

“…The purpose of this section is to generalise this result to left invariant Randers metrics on the 3-dimensional Heisenberg group. We note that S. Deng and Z. Hu have recently obtained some remarkable results for curvatures of homogeneous Randers metrics ( [5,6,9]). …”

On Flag Curvature of Homogeneous Randers Spaces

“…This results in a large numbers of examples of non-Riemannian homogeneous Randers spaces of Douglas type with negative flag curvature. In [HD11], we proved that a homogeneous Randers space with almost isotropic S-curvature and negative Ricci scalar must be Riemannian. The above argument shows that the restriction in S-curvature cannot be dropped.…”

“…For example, in [DE08] the second author obtained a very simple formula for S-curvature of homogeneous Randers spaces and gave some interesting applications of the formula. Based on this formula and the previous work of Berger, Wallach, Aloff-Wallach and Bérard-Bergery on homogeneous Riemannian manifolds of positive sectional curvature, we classified all the homogeneous Randers spaces with isotropic S-curvature and positive flag curvature in [HD11]. It seems hopeful that we can get a simple formula for flag curvature of an arbitrary homogeneous Randers spaces.…”