Orthogonality of Homogeneous geodesics on the tangent bundle

Let ξ = (G ×K G/K, ρξ,G/K, G/K) be the associated bundle and τG/K = (TG/K, πG/K,G/K,Rm) be the tangent bundle of special examples of odd dimension solvable Lie groups equipped with left invariant Riemannian metric. In this paper we prove some conditions about the existence of homogeneous geodesic on the base space of τG/K and homogeneous (geodesic) vectors on the fiber space of ξ .