On the structure of geodesic orbit Riemannian spaces
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.
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