Comptes Rendus
no. 11
Differential Geometry
Vector fields dynamics as geodesic motion on Lie groups

Presented by: Étienne Ghys

Gabriel Teodor Pripoae1
1 University of Bucharest, Faculty of Mathematics and Informatics, 14 Academiei St., Bucharest, Romania
Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 865-868.

We study to what extent vector fields on Lie groups may be considered as geodesic fields. For a given left invariant vector field on a Lie group, we prove there exists a Riemannian metric whose geodesics are its trajectories. When we consider left invariant metrics, differences between the Riemannian and the Lorentzian cases appear, coded by properties of the Lie algebra.

On étudie les conditions pour que les champs de vecteurs sur les groupes de Lie devienent des champs géodésiques. Pour un champ de vecteurs invariant à gauche, donné sur un groupe de Lie, on prouve qu'il existe une métrique riemannienne dont les géodésiques en sont les trajectoires. Dans le cas des métriques invariantes, on met en évidence certaines differences entre le cas riemannien et celui lorentzien, codées par des propriétés de l'algèbre de Lie.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.04.009

Gabriel Teodor Pripoae 1

1 University of Bucharest, Faculty of Mathematics and Informatics, 14 Academiei St., Bucharest, Romania 
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Gabriel Teodor Pripoae. Vector fields dynamics as geodesic motion on Lie groups. Comptes Rendus. Mathématique, Volume 342 (2006) no. 11, pp. 865-868. doi : 10.1016/j.crma.2006.04.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.009/

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