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 five‎. ‎Then we explicitly give the flag 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 five-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 five-dimensional two-step homogeneous Randers nilmanifolds of Douglas type are never weakly symmetric‎. ‎The existence 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 flat Finsler spaces are also obtained.