Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

TL;DR

This paper investigates geodesic orbit metrics on homogeneous spaces formed by classical Lie groups with specific subgroup structures, including Stiefel and Grassmann manifolds, focusing on cases where the subgroup is semisimple.

Contribution

It characterizes geodesic orbit metrics on a broad class of homogeneous spaces involving classical Lie groups and semisimple subgroups, extending understanding of g.o. spaces.

Findings

Classification of g.o. metrics on Stiefel and Grassmann manifolds

Identification of conditions for geodesic orbit property in these spaces

Extension of known results to new classes of homogeneous bundles

Abstract

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit metric. We study the geodesic orbit spaces of the form $(G/H,g)$, such that $G$ is one of the compact classical Lie groups $\SO(n)$, $U(n)$, and $H$ is a diagonally embedded product $H_1\times \cdots \times H_s$, where $H_j$ is of the same type as $G$. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces $(G/H,g)$ with $H$ semisimple.