Journal of Dynamical and Control Systems

, Volume 3, Issue 3, pp 391–418 | Cite as

Characterizations of hamiltonian geodesics in sub-riemannian geometry

  • M. Alcheikh
  • P. Orro
  • F. Pelletier
Article

Abstract

Let (M, ε, g) be a sub-Riemannian manifold. The geodesics of (M, ε, g) are either Hamiltonian or srictly abnormal. In this paper we give necessary and sufficient conditions for a geodesic curve to be Hamiltonian, and we apply this to the study of totally geodesic submanifolds of Lie groups. We also give an example of a foliation by totally geodesic submanifolds, each leaf containing a one parameter family of geodesics which are Hamiltonian in the leaf but stricly abnormal in the ambient space.

1991 Mathematics Subject Classification

58A30 53C22 93B29 49K15 

Key words and phrases

Sub-Riemannian geometry distributions Hamiltonian geodesic abnormal geodesic nilpotent Lie groups 

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Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • M. Alcheikh
    • 1
  • P. Orro
    • 1
  • F. Pelletier
    • 1
  1. 1.Laboratoire de MathématiquesUniversité de SavoieLe Bourget-du-Lac, CedexFrance

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