Abstract

In this paper, firstly we study geodesic vectors for the $m$-th root homogeneous Finsler space admitting $(\alpha,\beta)$-type. Then we obtain the necessary and sufficient condition for an arbitrary non-zero vector to be a geodesic vector for the $m$-th root homogeneous Finsler metric under mild conditions. Finally, we consider a quartic homogeneous Finsler metric on a simply connected nilmanifold of dimension five equipped with an invariant Riemannian metric and an invariant vector field. We study its geodesic vectors and classify the set of all the homogeneous geodesics on $5$-dimensional nilmanifolds.

Keywords: Geodesic vectors; $m$-th root Finsler metric; quartic Finsler metric; nilpotent Lie groups; invariant metric; nilmanifolds.

MSC: 22E60, 53C30, 53C60

DOI: 10.57016/MV-YTVW3833

Pages:  1$-$14