Geodesic
Invariant totally geodesic unit vector fields on three-dimensional Lie groups
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (2007) no. 2, pp. 253-276 
A. Yampolsky. Invariant totally geodesic unit vector fields on three-dimensional Lie groups. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (2007) no. 2, pp. 253-276. http://geodesic.mathdoc.fr/item/JMAG_2007_3_2_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We give a complete list of left-invariant unit vector fields on three-dimensional Lie groups equipped with a left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group equipped with the Sasaki metric. As a result we obtain that each three-dimensional Lie group admits totally geodesic unit vector field under some conditions on structural constants. From a geometrical viewpoint, the field is either parallel or a characteristic vector field of a natural almost contact structure on the group.

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