Homogeneous Randers Spaces

Abstract

In this chapter we study homogeneous Randers spaces. In Sect. 7.1, we give a method to construct invariant Randers metrics on a coset space of a Lie group, through studying invariant vector fields on homogeneous manifolds. In Sect. 7.2, we use the formula of the Levi-Civita connection of an invariant Riemannian metric on a coset space to deduce an explicit simple formula for the S-curvature of invariant Randers metrics. In Sects. 7.3 and 7.4, we study homogeneous Einstein–Randers metrics. The main results are a rigidity result that every homogeneous Einstein–Randers metric with negative Ricci scalar must be Riemannian, and a complete classification of all homogeneous Einstein–Randers metrics on spheres. As a result, we find many new homogeneous Einstein metrics on spheres. In Sect. 7.5, we prove that a homogeneous Randers metric is Ricci quadratic if and only if it is Berwald. Finally, in Sects. 7.6 and 7.7, we study homogeneous Randers metrics with positive flag curvature or negative flag curvature. This results in an isometric classification of homogeneous Randers spaces with positive flag curvature and almost isotropic S-curvature, and a rigidity result asserting that a homogeneous Randers metric with negative flag curvature and almost isotropic S-curvature must be Riemannian.