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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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