On the structure of geodesic orbit Riemannian spaces
type: Item
instance of: scholarly article
title: On the structure of geodesic orbit Riemannian spaces
zbMATH Open document ID: 1381.53088
published in: Annals of Global Analysis and Geometry
publication date: 2017-10-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1611.01050
review text: Riemannian homogeneous spaces \(M=G/H\), where the stationary subgroup \(H\) in a Lie group \(G\) is compact, are considered. Such \(M\) is called a space with homogeneous geodesics or geodesic orbit space (GO-space), if any geodesic of M is an orbit of a 1-parameter subgroup of the Lie group G. GO-spaces may be considered as a natural generalization of Riemannian symmetric spaces. But the class of GO-spaces is much larger than the class of symmetric spaces. For example, any homogeneous space \(G/H\) of a compact Lie group \(G\) admits a Riemannian metric \(g\) such that \((G/H,g)\) is a GO-space. In this article, some general properties of Riemannian homogeneous spaces \(G/H\) are proved. Then, as a main result, the structures of the nilradical and of the radical of the Lie algebra of the isometry group of GO-spaces are described. Also some new tools to study GO-spaces, related to compact Lie group representations with non-trivial principal isotropy algebras, are given. Some new examples of homogeneous Riemannian GO-spaces and some unsolved questions about them are presented.
reviewed by: V. V. Gorbatsevich
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 6798473
zbMATH Keywords: homogeneous Riemannian manifold
zbMATH Keywords: symmetric space
zbMATH Keywords: geodesic orbit Riemannian space
Wikidata QID: Q115384565
author: Yurii G. Nikonorov
MaRDI profile type: Publication
OpenAlex ID: W3098785015
arXiv ID: 1611.01050
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DOI: 10.1007/S10455-017-9558-0