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Geodesic orbit Riemannian structures on \(\mathbf{R}^n\). (English) Zbl 1407.53032

A complete Riemannian manifold \((M,g)\) is said to be a geodesic orbit manifold (GO-manifold) if every geodesic of \(M\) is an orbit of a one-parameter group of isometries. GO-manifolds are necessarily homogeneous. The aim of the present paper is to study both the geometry and the symmetry properties of GO-manifolds that are diffeomorphic to \(\mathbb{R}^n\).
The main result of the paper is as follows: If \((M,g)\) is a GO-manifold diffeomorphic to \(\mathbb{R}^n\), then
(1)
\((M,g)\) is the total space of a Riemannian submersion \(\pi:M\longrightarrow P\), where the base space \(P\) is a Riemannian symmetric space of noncompact type. The fibers are totally geodesic and are isometric to a GO-nilmanifold \((N,g)\) of step size at most two,
(2)
\(M\) admits a simply-transitive solvable group of isometries of the form \(S\times N\), where \(S\) is an Iwasawa subgroup of a semisimple Lie group and \(N\) is the group in (1).
Along the way, various structural properties of more general GO-manifolds are established.

MSC:

53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
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