On 5-dimensional 2-step homogeneous randers nilmanifolds of Douglas type

In this paper we first obtain the non-Riemannian Randers metrics of Douglas type on two-step homogeneous nilmanifolds of dimension fiveý. ýThen we explicitly give the flag curvature formulae and the S-curvature formulae for the Randers metrics of Douglas type on these spacesý. ýMoreoverý, ýwe prove that the only simply connected five-dimensional two-step homogeneous Randers nilmanifolds of Douglas type which are Ricci-quadratic have a three-dimensional centreý. ýWe also prove that all simply connected five-dimensional two-step homogeneous Randers nilmanifolds of Douglas type are never weakly symmetricý. ýThe existence of homogeneous Randers spaces of Douglas type with vanishing S -curvature which are never g.oý. ýFinsler spaces is also proved and some examples of locally projectively flat Finsler spaces are also obtained.