Homogeneous spaces with geodesic orbit Riemannian metrics and with integrable invariant distributions

TL;DR

This paper investigates homogeneous spaces with specific geometric properties, distinguishing those with integrable invariant distributions from those with geodesic orbit metrics, and identifies examples with Einstein metrics.

Contribution

It characterizes homogeneous spaces with geodesic orbit metrics and finds examples with integrable distributions that do not admit such metrics.

Findings

Identified homogeneous spaces with integrable invariant distributions not supporting geodesic orbit metrics.

Discovered several spaces that admit invariant Einstein metrics.

Provided examples differentiating the two types of homogeneous spaces.

Abstract

We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary invariant Riemannian metric on the space, every geodesic is an orbit of a 1-parameter subgroup of the isometry group. We found several homogeneous spaces of the first type that are not spaces of the second type. Among them there are several homogeneous spaces that admit invariant Einstein metrics.