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On the structure of geodesic orbit Riemannian spaces

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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.

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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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  1. Southern Mathematical Institute of VSC RAS, Markus str., 22, Vladikavkaz, Russia, 362027

    Yuriĭ Gennadievich Nikonorov

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Nikonorov, Y.G. On the structure of geodesic orbit Riemannian spaces. Ann Glob Anal Geom 52, 289–311 (2017). https://doi.org/10.1007/s10455-017-9558-0

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