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On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. (English) Zbl 0980.53061
Geom. Dedicata 81, No. 1-3, 209-214 (2000); erratum ibid. 84, No. 1-3, 331-332 (2001).
Let $$M=G/H$$, $$G$$ a connected Lie group, $$H\subset G$$ a closed subgroup, $$\varphi:G \times M\to M$$ the canonical left action. If $$\nabla$$ is an affine connection on $$M$$ which is invariant by $$\varphi$$ then a geodesic $$\gamma$$ of $$\nabla$$ is called homogeneous if it coincides with a 1-parameter subgroup, $$\gamma(t)=\varphi (\exp tx,z)$$, $$t\in\mathbb{R}$$, $$x\in{\mathfrak g}=\text{Lie} G$$, $$z=\gamma(0)$$.
In this paper the authors study the existence of homogeneous geodesics, on homogeneous Riemannian manifolds. The main results obtained are Proposition 3, where they show that on a homogeneous Riemannian manifold $$M$$ either there exists a homogeneous geodesic through any point $$o\in M$$ or $$M=G/H$$ where $$G$$ is a semisimple group of isometries, and Theorem 1, where they show that in the last case there always exist $$m=\dim M$$, mutually orthogonal homogeneous geodesics. Moreover, as a consequence of Theorem 2 they show that every isotropy irreducible homogeneous Riemannian $$G/H$$, $$G$$ semisimple, is a naturally reductive space.

##### MSC:
 53C30 Differential geometry of homogeneous manifolds 53C22 Geodesics in global differential geometry
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