On the Finsler Geometry of Four-Dimensional Einstein Lie Groups

Abstract

In this paper, we study left invariant $$(\\alpha ,\\beta )$$\n -metrics on four-dimensional real Lie groups equipped with left invariant Einstein Riemannian metrics. We classify all left invariant $$(\\alpha ,\\beta )$$\n -metrics of Berwald type induced by a left invariant Einstein Riemannian metric and a left invariant vector field and show that all of them are locally Minkowskian. \nAll left invariant Randers metrics of Douglas type, and all Einstein Kropina metrics induced by a left invariant Riemannian metric and a left invariant vector field, \nare classified. Finally, the flag curvatures of these spaces are investigated and in a special case the geodesics are computed.