Geodesic orbit spaces of compact Lie groups of rank two

TL;DR

This paper classifies simply connected geodesic orbit spaces with compact Lie groups of rank two, showing that most are spheres and projective spaces with metrics from Hopf fibrations, except for those with bi-invariant metrics.

Contribution

It provides a complete classification of geodesic orbit spaces for rank two compact Lie groups, highlighting the special role of Hopf fibration induced metrics.

Findings

Most geodesic orbit spaces are spheres and projective spaces.

Metrics not from bi-invariant metrics are characterized by Hopf fibrations.

Classification is complete for simply connected spaces with rank two groups.

Abstract

Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H,g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.