Constant Flag Curvature Randers Metrics

One class of the most important and the simplest Finsler spaces is the Randers spaces. They are more closely related to Riemanian spaces than any other Finsler spaces. Randers metric was first introduced by Randers in 1941, when he studied the metric problems in the 4-space of general relativity. The flag curvature of Finsler spaces is a basic problem in Finsler geometry. In this paper, we first give the formula of the flag curvature of some invariant Randers metric on homogeneous Riemannian manifolds, and then find the flag curvature of some Randers metric on S~2×S~1 is always positive; Finally we generalize the formula to generalized Heisenberg groups.In the preface, we introduce the development of Finsler Geometry, the direction of our research and the construction of this paper.In charpter1, we give some basic preliminaries, definitions and basic results in Finsler and Randers spaces, and give some preliminaries of generalized Heisenberg groups.In charpter2, we first give the formula of the flag curvature of the invariant Randers metrics on homogeneous Riemanian spaces, and find the flag curvature of some Randers metric on S~2×S~1 is always positive; Then we generalize it to generalized Heisenberg groups, and find on some special flags, the sectional curvature and the flag curvature have some special relationships; Finally, we give the formula of the cartan curvature of the invariant Randers metrics on homogeneous Riemanian spaces.