On homogeneous geodesics and weakly symmetric spaces

TL;DR

This paper provides new insights into the conditions under which geodesics are homogeneous in Riemannian manifolds, proving that weakly symmetric spaces are geodesic orbit spaces and exploring properties of homogeneous geodesics.

Contribution

It establishes a sufficient condition for geodesics to be homogeneous and offers a new proof that weakly symmetric spaces are geodesic orbit manifolds, expanding understanding of homogeneous geodesics.

Findings

Every weakly symmetric space is a geodesic orbit manifold.

Examples of homogeneous geodesics with torus closures of dimension ≥ 2.

Analysis of homogeneous geodesics in Lie groups with left-invariant metrics.

Abstract

In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an $1$-parameter isometry group. As an application of this result, we provide a new proof of the fact that every weakly symmetric space is geodesic orbit manifold, i.e. all its geodesics are homogeneous. We also study general properties of homogeneous geodesics, in particular, the structure of the closure of a given homogeneous geodesic. We present several examples where this closure is a torus of dimension $\geq 2$ which is (respectively, is not) totally geodesic in the ambient manifold. Finally, we discuss homogeneous geodesics in Lie groups supplied with left-invariant Riemannian metrics.