Orthogonality of Homogeneous geodesics on the tangent bundle

TL;DR

This paper investigates the conditions for the existence of homogeneous geodesics on certain tangent bundles of solvable Lie groups with invariant metrics, focusing on their orthogonality properties.

Contribution

It establishes new conditions for the existence of homogeneous geodesics and vectors on tangent bundles of specific solvable Lie groups.

Findings

Conditions for homogeneous geodesic existence on base space

Criteria for homogeneous geodesic vectors on fiber space

Orthogonality properties of these geodesics

Abstract

Let $\xi=(G\times_{K} \mathcal{G} / \mathcal{K}, \rho_{\xi}, \emph{G} / \emph{K},\mathcal{G} / \mathcal{K})$ be the associated bundle and $\tau_{G/K}=(T_{G/K},\pi_{G/K},G/K, \textrm{R}^{m})$ 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 $\tau_{G/K}$ and homogeneous (geodesic) vectors on the fiber space of $\xi$ .