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

Document Type : Original Article


1 Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, 81746-73441-Iran. Email:

2 Department of Mathematics, Isfahan University of Technology, Isfahan, 84156-83111-Iran.

3 Faculty of Basic Sciences, University of Shahreza, P. O. Box: 86149- 56841, Shahreza, Iran. Email:


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.