## 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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