Classification of Douglas $(\alpha ,\beta )$-metrics on five dimensional nilpotent Lie groups

Abstract

In this paper we classify all simply connected five dimensional nilpotent Lie groups which admit $(\alpha ,\beta )$-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and $S$-curvature.