On the geodesics of a homogeneous Finsler space with a special (α, β)−metric

One of the most important concepts in geometry is of geodesics. Geodesic in a manifold is the generalization of notion of a straight line in an Euclidean space. A geodesic in a homogeneous Finsler space (G/H, F) is called homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. Homogeneous geodesics on homogeneous Riemannian manifolds have been studied by many authors. Latifi has extended the concept of homogeneous geodesics in homogeneous Finsler spaces. He has given a criterion for characterization of geodesic vectors. Latifi and Razavi have studied homogeneous geodesics in a 3-dimensional connected Lie group with a left invariant Randers metric and show that all the geodesics on spaces equipped with such metrics are homogeneous.  In this paper, first we give basic definitions required to define a honogeneous Finsler space. Next, we study geodesics and geodesic vectors for homogeneous Finsler space with infinite series (α, β)-metric. Next, we give a lemma in which the existence of invariant vector field corresponding to 1-form β for a homogeneous Finsler space with infinite series metric is proved. Further, we find necessary and sufficient condition for a non-zero vector in this homogeneous space to be a geodesic vector.