Cyclic Finser metrics on homogeneous spaces

In this paper, we generalize the notion of cyclic metric to homogeneous Finsler geometry. Firstly, we prove that a homogeneous Finsler space $(G/H, F)$ must be symmetric when it satisfies the naturally reductive and cyclic conditions simultaneously. Then we prove that a Finsler cyclic Lie group which is either flat or nilpotent must have an Abelian Lie algebra. Finally, we show how to induce a cyclic $(\alpha,\beta)$ metric from a cyclic Riemannian metric. Using this method, we construct a Randers cyclic Lie group.