On 5-dimensional 2-step homogeneous randers nilmanifolds of Douglas type
2017
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.
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