Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

TL;DR

This paper investigates geodesic orbit metrics on certain homogeneous spaces, including quaternionic Stiefel and flag manifolds, focusing on spaces where the isotropy subgroup is semisimple, expanding understanding of their geometric structures.

Contribution

It characterizes geodesic orbit metrics on quaternionic homogeneous bundles over Stiefel manifolds with semisimple isotropy, broadening the classification of such spaces.

Findings

Identification of conditions for g.o. metrics on quaternionic Stiefel manifolds

Extension of g.o. space classification to new homogeneous bundles

Analysis of the geometric properties of these spaces

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 $(\Sp(n)/\Sp(n_1)\times \cdots \times \Sp(n_s), g)$, with $0<n_1+\cdots +n_s\leq n$. Such spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces $(G/H,g)$ with $H$ semisimple.