Geodesic
Homogeneous Geodesics in 3-dimensional Homogeneous Affine Manifolds
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 50 (2011) no. 1, pp. 29-42.

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For studying homogeneous geodesics in Riemannian and pseudo-Riemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which works, at least potentially, in every given case. In the affine differential geometry, there is not such a universal formula. In the previous work, we proposed a simple method of investigation of homogeneous geodesics in homogeneous affine manifolds in dimension 2. In the present paper, we use this method on certain classes of homogeneous connections on the examples of 3-dimensional Lie groups.
Classification : 53B05, 53C30
Keywords: affine connection; affine Killing vector field; homogeneous manifold; homogeneous geodesic 
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Dušek, Zdeněk; Kowalski, Oldřich; Vlášek, Zdeněk. Homogeneous Geodesics in 3-dimensional Homogeneous Affine Manifolds. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 50 (2011) no. 1, pp. 29-42. http://geodesic.mathdoc.fr/item/AUPO_2011__50_1_a3/