On the structure of geodesic orbit Riemannian spaces

TL;DR

This paper investigates the structure and properties of geodesic orbit Riemannian spaces, focusing on their submanifolds, Lie algebra components, and new tools for analysis, including examples and open questions.

Contribution

It introduces new methods and examples for studying geodesic orbit Riemannian spaces, and analyzes their Lie algebra structures and submanifold properties.

Findings

Characterization of totally geodesic submanifolds as geodesic orbit spaces

Description of the nilradical and radical structures in the isometry Lie algebra

Introduction of new tools involving compact Lie group representations

Abstract

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic orbit Riemannian space, we describe the structure of the nilradical and the radical of the Lie algebra of the isometry group. In the final part, we discuss some new tools to study geodesic orbit Riemannian spaces, related to compact Lie group representations with non-trivial principal isotropy algebras. We discuss also some new examples of geodesic orbit Riemannian spaces, new methods to obtain such examples, and some unsolved questions.