Geodesic orbit spaces in real flag manifolds

TL;DR

This paper classifies geodesic orbit metrics on real flag manifolds, revealing more cases of such metrics due to the complex structure of the isotropy representation compared to the complex case.

Contribution

It provides a classification of g.o. metrics on real flag manifolds, highlighting the impact of equivalent submodules in the isotropy representation.

Findings

More g.o. metrics exist on real flag manifolds than in the complex case.

Invariant metrics depend on more parameters due to equivalent submodules.

The classification includes new non-trivial g.o. metrics.

Abstract

We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find more cases in which non-trivial g.o. metrics exist.