Journal of Dynamical and Control Systems

, Volume 2, Issue 3, pp 321–358 | Cite as

Exponential mappings for contact sub-Riemannian structures

  • A. A. Agrachev
Article

Abstract

On sub-Riemannian manifolds any neighborhood of any point contains geodesics which are not length minimizers; the closures of the cut and the conjugate loci of any pointq containq. We study this phenomenon in the case of a contact distribution, essentially in the lowest possible dimension 3, where we extract differential invariants related to the singularities of the cut and the conjugate loci nearq and give a generic classification of these singularities.

1991 Mathematics Subject Classification

53B 49L05 

Key words and phrases

Sub-Riemannian geometry contact structure differential invariants conjugate locus dut locus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. A. Agrachev, Methods of control theory in nonholonomic geometry.Proc. ICM-94, Zürich, Birkhäuser, 1995, 1473–1483.Google Scholar
  2. 2.
    A. A. Agrachev, El-H. C. El Alaoui, J. -P. Gauthier, and I. Kupka, Generic sub-Riemannian metrics onR 3 Compt. Rend. Acad. Sci. Ser. 1 322 (1996), 377–384.Google Scholar
  3. 3.
    A. A. Agrachev and R. V. Gamkrelidze, Exponential representation of flows and chronological calculus. (Russian)Mat. sb. 107 (1978) 467–532. (English translation: Math. USSR Sb.29 (1979), 727–785.)Google Scholar
  4. 4.
    A. A. Agrachev, Symplectic methods for optimization and control (to appear in: Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek. Eds,Marcel Dekker).Google Scholar
  5. 5.
    A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems.Acta Appl. Math. 14 (1989), 191–237.Google Scholar
  6. 6.
    A. A. Agrachev, S. Stefani, and P. L. Zezza, Strong minima in optimal control (in preparation).Google Scholar
  7. 7.
    V. I. Arnold, Mathematical methods in classical mechanics, Third edition,Nauka, Moscow, 1989.Google Scholar
  8. 8.
    R. W. Brockett, Control theory and singular Riemannian geometry. In: New Directions in Applied Mathematics, P. J. Hilton and G. S. Young, Eds.Springer Verlag, 1981.Google Scholar
  9. 9.
    Ge Zhong, Horizontal path space and Carnot-Caratheodory metrics.Pac. J. Math. 161, (1993), 255–286.Google Scholar
  10. 10.
    M. Golubitsky and V. Guillemin, Stable mappings and their singularities.Springer Verlag, New York, 1973.Google Scholar
  11. 11.
    M. Gromov, Carnot-Caratheodory spaces seen from within.Preprint IHES/M/94/6, 1994Google Scholar
  12. 12.
    R. Montgomery, The isoholonomic problem and some applications,Commun. Math. Phys. 128, (1990), 565–592.Google Scholar
  13. 13.
    A. V. Sarychev, The index of the second variation of a control system.Mat. Sb. 113 (1980) 464–486. (English translation:Math. USSR Sb. 41 (1982) 383–401.)Google Scholar
  14. 14.
    A. M. Vershik and V. Y. Gershkovich, Nonholonomic dynamical systems. Geometry of distributions and variational problems. (Russian) In: Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamentalnye Napravleniya, Vol. 16,VINITI, Moscow, 1987, 5–85. (English translation) in:Encyclopedia of Math. Sci. 16, Dynamical Systems 7,Springer Verlag).Google Scholar
  15. 15.
    H. Whitney, On singularities of mappings of Euclidean spaces.Ann. math. 62 (1955).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • A. A. Agrachev
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations