Geodesic
Homogeneous geodesics in a three-dimensional Lie group
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 261-270.

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O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.e\. one geodesic which is an orbit of a one-parameter group of isometries. In [KNV] the related two problems were studied and a negative answer was given to both ones: (1) Let $M=K/H$ be a homogeneous Riemannian manifold where $K$ is the largest connected group of isometries and $\dim M\geq 3$. Does $M$ always admit more than one homogeneous geodesic? (2) Suppose that $M=K/H$ admits $m = \dim M$ linearly independent homogeneous geodesics through the origin $o$. Does it admit $m$ mutually orthogonal homogeneous geodesics? In this paper the author continues this study in a three-dimensional connected Lie group $G$ equipped with a left invariant Riemannian metric and investigates the set of all homogeneous geodesics.
Classification : 53C20, 53C22, 53C30
Keywords: Riemannian manifold; homogeneous space; geodesics as orbits 
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Marinosci, Rosa Anna. Homogeneous geodesics in a three-dimensional Lie group. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 261-270. http://geodesic.mathdoc.fr/item/CMUC_2002__43_2_a5/