Hamid Reza Salimi Moghaddam - Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, ۸۱۷۴۶-۷۳۴۴۱-Iran. Email: salimi.moghaddam@gmail.com
Hossein Abedi Karimi - Department of Mathematics, Isfahan University of Technology, Isfahan, ۸۴۱۵۶-۸۳۱۱۱-Iran. Email:hossein.abedikarimi@gmail.com
Mehri Nasehi - Faculty of Basic Sciences, University of Shahreza, P. O. Box: ۸۶۱۴۹- ۵۶۸۴۱, Shahreza, Iran. Email: nasehi.mehri@gmail.com

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.

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