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

In this paper we rst obtain the non-Riemannian Randers metrics of Douglas type on two-step homogeneous nilmanifolds of di- mension ve. Then we explicitly give the ag curvature formulae and the S-curvature formulae for the Randers metrics of Douglas type on these spaces. Moreover, we prove that the only simply connected ve- dimensional two-step homogeneous Randers nilmanifolds of Douglas type which are Ricci-quadratic have a three-dimensional centre. We also prove that all simply connected ve-dimensional two-step homogeneous Ran- ders nilmanifolds of Douglas type are never weakly symmetric. The ex- istence of homogeneous Randers spaces of Douglas type with vanishing S-curvature which are never g.o. Finsler spaces is also proved and some examples of locally projectively at Finsler spaces are also obtained.