Homogeneous geodesics on the solvable homogeneous principal bundles

Abstract

Let $M=G/K$ be a homogeneous differentiable manifold .We consider the
homogeneous bundle $\Im=(G,\pi,G/K,K)$
and the tangent bundle $\tau_{G/K}$ of $M=G/K$. Let $G$ be a connected Lie group and $K$
a closed subgroup of $G$. We take the Lie algebras $\cal{G}$ and
$\cal{K}$ of $G$ and $K$ respectively and define $\xi=(G\times_{K}
\cal{ G} / \cal {K}, \rho_{\xi}, \emph{G} / \emph{K},\cal {G} /
\cal {K})$ to be the bundle associated with $\Im$. We consider
that $ \xi$ is strongly isomorphic to tangent bundle
$\tau_{G/K}=(T_{G/K},\pi_{G/K},G/K, {\textbf{R}}^{m})$ .\\

In this paper we take $G$ be a solvable Lie group and prove some
results about the existence of homogeneous geodesics and geodesic
vectors on the fiber