## On a class of geodesic orbit spaces with abelian isotropy subgroup.(English)Zbl 1478.53095

We say that a Riemannian manifold $$(M,g)$$ is a geodesic orbit (g.o) manifold if any geodesic of $$M$$ is an orbit of a one-parameter subgroup of the entire isometry group of $$M$$. The author considers g.o. spaces of the form $$(G/S, g)$$ such that $$G$$ is a compact, connected, semisimple Lie group and the isotropy subgroup $$S$$ is abelian. He states (Theorem 1.1) that $$G/S$$ is a g.o space if and only if $$g$$ is a $$G$$-naturally reductive metric, i.e., a suitable $$G$$-invariant metric (resp., an induced from a bi-invariant Riemannian metric on $$G$$).

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds
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### References:

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