On the structure of geodesic orbit Riemannian spaces

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.

20 pages, improved Section 2, small corrections, comments are welcome

References in corpus (2)

• Compact Riemannian Manifolds with Homogeneous Geodesics
• Invariant Einstein metrics on Ledger-Obata spaces

Cited by in corpus (12)

• Geodesic Orbit Riemannian Structures on ${\mathbf{R}}^n$
• On left-invariant Einstein Riemannian metrics that are not geodesic orbit
• Geodesic orbit Riemannian spaces with two isotropy summands. I
• On invariant Riemannian metrics on Ledger-Obata spaces
• Compact geodesic orbit spaces with a simple isotropy group
• Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds
• Spectral properties of Killing vector fields of constant length
• On homogeneous geodesics and weakly symmetric spaces
• Riemannian $M$-spaces with homogeneous geodesics
• Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups
• On geodesic orbit nilmanifolds
• Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds