On the Finsler Geometry of Four-Dimensional Einstein Lie Groups

Abstract

In this paper, we study left invariant $(\alpha ,\beta )$-metrics on four-dimensional real Lie groups equipped with left invariant Einstein Riemannian metrics. We classify all left invariant $(\alpha ,\beta )$-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. All 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, are classified. Finally, the flag curvatures of these spaces are investigated and in a special case the geodesics are computed.