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

In this paper, we give a classification of left-invariant Douglas and Berwald (α,β)-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.

Keywords:

nilpotent Lie group , Riemannian metric , (α , β)-metric , flag curvature