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Annals of Global Analysis and Geometry

, Volume 52, Issue 3, pp 289–311 | Cite as

On the structure of geodesic orbit Riemannian spaces

  • Yuriĭ Gennadievich Nikonorov
Article

Abstract

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterized 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.

Keywords

Homogeneous Riemannian manifolds Symmetric spaces Homogeneous spaces Geodesic orbit Riemannian spaces 

Mathematics Subject Classification

53C30 (primary) 53C20 53C25 53C35 (secondary) 

Notes

Acknowledgements

The author is indebted to Prof. Valerii Berestovskii, to Prof. Carolyn Gordon, and to Prof. Èrnest Vinberg for helpful discussions concerning this paper. The author is grateful to the anonymous referee for helpful comments and suggestions that improved the presentation of this paper.

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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Southern Mathematical Institute of VSC RASVladikavkazRussia

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