Douglas (α,β)-metrics on four-dimensional nilpotent Lie groups

In this paper, we give a classification of left-invariant Douglas and Berwald $(alpha,beta)$-metrics on simply connected four-dimensional nilpotent Lie groups. We show that there are not any bi-invariant Randers metrics on four-dimensional nilpotent Lie groups. Then, we explicitly give the flag curvature formulas and geodesic vectors of these spaces. Finally, we give the formula of $S$-curvature of left-invariant Randers metrics of Douglas type.