Inventiones mathematicae

, Volume 114, Issue 1, pp 435–461

Rigidity of integral curves of rank 2 distributions

  • Robert L. Bryant
  • Lucas Hsu
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Second Edition. Graduate Texts in Mathematics60, Springer, New York, 1989Google Scholar
  2. Bliss, G.: The problem of Lagrange in the calculus of variations. Am. J. Math.52, 673–744 (1930)Google Scholar
  3. Brockett, R., Dai, L.: Non-holonomic kinematics and the role of elliptic functions in constructive controlability (1992, preprint)Google Scholar
  4. Bryant, R., Chern, S.-S., Gardner, R., Goldschmidt, H., Griffiths, P.A.: Exterior Differential Systems. MSRI Publications18, Springer, New York, 1991Google Scholar
  5. Cartan, É.: Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Ec. Norm.27, 109–192 (1910)Google Scholar
  6. Cartan, É.: Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes. Bull. Soc. Math. France42, 12–48 (1914)Google Scholar
  7. Cartan, É.: Sur l'intégration de certains systèmes indéterminés d'équations différentielles. J. Reine Angew. Math.145, 86–91 (1915)Google Scholar
  8. Cartan, É.: Leçons sur les Invariants Intégraux. Hermann, Paris, 1924Google Scholar
  9. Chow, W.L.: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann.117, 98–105 (1939)Google Scholar
  10. Gardner, R.: Invariants of Pfaffian systems. Trans. Am. Math. Soc.126, 514–533 (1967)Google Scholar
  11. Giaro, A., Kumpera, A., Ruiz, C.: Sur la lecture correcte d'un résult d'Élie Cartan. C. R. Acad. Sc.287 Série A, 241–244 (1978)Google Scholar
  12. Goursat, E.: Leçons sur le problem de Pfaff. Hermann, Paris, 1923Google Scholar
  13. Griffiths, P.A.: Exterior Differential Systems and the Calculus of Variations. Progr. Math.25, Birkhäuser, Boston, 1983Google Scholar
  14. Gromov, M.: Partial Differential Relations. Springer, Berlin Heidelberg, 1986Google Scholar
  15. Hamenstädt, U.: Some regularity theorems for Carnot-Carathéodory metrics. J. Differ. Geom.,32, 819–850 (1990)Google Scholar
  16. Hermann, R.: Differential Geometry and the Calculus of Variations. Math. Sci. Eng.49, Academic Press, New York 1968Google Scholar
  17. Hilbert, D.: Über den Begriff der Klasse von differentialgleichungen. Math. Ann.73, 95–108 (1912)Google Scholar
  18. hsu, L.: Calculus of Variations via the Griffiths formalism. J. Differ. Geom.36, 551–589 (1992)Google Scholar
  19. Montgomery, R.: A counterexample in subRiemannian geometry (preprint. 1993)Google Scholar
  20. Mūto, Y.: Critical curves on a two-dimensional distribution. Tensor, N.S.25, 337–352 (1972)Google Scholar
  21. Pansu, P.: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math.129, 1–60 (1989)Google Scholar
  22. Pontrjagin, L., Boltyanskii, V., Gamkredlidze, R., Mishchenko, E.: The Mathematical Theory of Optimal Processes. Wiley Interscience, New York, 1962Google Scholar
  23. Rayner, C.: The exponential map for the Lagrange problem on differentiable manifolds. Phil. Trans. of the Royal Soc. of London, ser. A, Math. and Phys. Sci., no. 1127,262, 299–344 (1967)Google Scholar
  24. Sluis, W.: Absolute Equivalence and its Applications to Control Theory, a thesis presented to the University of Waterloo, Ontario, Canada, 1992Google Scholar
  25. Strichartz, R.: Sub-Riemannian geometry. J. Differ. Geom.24, 221–263 (1986)Google Scholar
  26. Strichartz, R.: Corrections to “Sub-Riemannian geometry”. J. Differ. Geom.30, 595–596 (1989)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Robert L. Bryant
    • 1
  • Lucas Hsu
    • 2
  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.School of MathematicsThe Institute for Advanced StudyPrincetonUSA

Personalised recommendations