Two-step homogeneous geodesics in homogeneous spaces

TL;DR

This paper investigates a new class of geodesics in homogeneous spaces, called two-step homogeneous geodesics, providing conditions for their existence and examples where all geodesics are of this form.

Contribution

It introduces the concept of two-step homogeneous geodesics, establishes conditions for their existence, and identifies classes of spaces where all geodesics are of this type.

Findings

Conditions for the existence of two-step homogeneous geodesics.

Examples of spaces where all geodesics are two-step homogeneous.

Application to spaces with invariant metrics and Riemannian submersions.

Abstract

We study geodesics of the form $\gamma(t)=\pi(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\pi:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of $G$ (i.e. $\gamma(t)=\pi(\exp (tX))$, $X\in \fr{g}$). We obtain sufficient conditions on a homogeneous space implying the existence of such geodesics for $X,Y\in \fr{m}=T_o(G/K)$. We use these conditions to obtain examples of Riemannian homogeneous spaces $G/K$ so that all geodesics of $G/K$ are of the above form. These include total spaces of homogeneous Riemannian submersions endowed with one parameter families of fiber bundle metrics, Lie groups endowed with special one parameter families of left-invariant metrics, generalised Wallach spaces, generalized flag manifolds, and $k$-symmetric spaces with $k$-even, equipped with certain one-parameter families of invariant metrics.