Invariant Einstein-Randers Spaces

The study of Einstein metrics in Riemannian geometry and, in a more gen-eral setting, Finsler geometry, is one of the fundamental problems in differential geometry. In this paper, we will study the invariant Einstein-Randers metrics on homogeneous spaces and Lie groups. This article involves the following three cases: (1) The invariant Einstein-Randers metrics on spheres; (2) The invariant Einstein-Randers metrics on the homogeneous spaces whose rank are more than 1; (3) The left-invariant Einstein-Randers metrics compact Lie groups.The Randers metrics have closed relations to the Riemannian metrics, and they can be related to the Riemannian metrics and its infinitesimal homothety by the nav igation data. Therefore. the existences of the Einstein-Randers metrics on Finsler manifolds have closed relations to the existences of the Einstein-Riemannian metrics on the corresponding Riemannian manifolds. We will study the invariant Einstein-Randers metrics on Finsler manifolds by the considering Einstein-Riemannian met-rics and the corresponding infinitesimal homothety on homogeneous spaces and com-pact Lie groups.In this article, we first study the invariant Einstein-Randers metrics on spheres. First of all, we study whether there are some non-zero invariant vector fields on spheres by the isotropy representation of them, and then, construct the invariant Einstein-Randers metrics on spheres, using the invariant vector fields and the invari-ant Einstein-Riemannian metrics on spheres Secondly, we prove that there no other invariant Einstein-Randers metrics on spheres, except the above metrics. Then we classified the metrics above by the way in Lie groups and obtain the full isome-try group of them. Therefore, we have obtained a complete description of all the invariant Einstein-Randers metrics on spheres. Using the method above, we also study the invariant Einstein-Randers metrics on some homogeneous spaces whose rank are more than 1. These metrics and the invariant Einstein-Randers metrics on spheres present the examples of the Einstein-Randers metrics whose flag curvatures are not of constant. Therefore, we have filled the gap that most of these metrics obtained before are of constant flag curvature.In fact, this method can be generalized to Lie groups. We obtain some left-invariant Einstein-Randers metrics on compact Lie groups, and we prove except these metrics there are no other such metrics. Then we give the formula of the geodesic through the identity of the group and one simple rigidity conclusion.When the group is simple. The above metrics have excellent geometrical prop-ertv. Moreover, we classified such metrics and describe the identity component of the isometry group. Later, we give the formula of the flag curva.ture of left-invariant Einstein-Randers metrics on the compact Lie group. Especially, we prove that these metrics have constant flag curvature if and only if they have scalar curvature, using the formulas.By studying the invariant Einstein-Randers metrics on homogeneous spaces and Lie groups, we obtain a large number of examples of invariant Einstein-Randers metrics whose flag curvature are not of constant, which are not Riemannian and have have special geometrical properties.