Journal of Mathematical Sciences

, Volume 83, Issue 4, pp 461–476 | Cite as

The tangent space in sub-riemannian geometry

  • A. Bellaïche


Vector Field Singular Point Tangent Space Heisenberg Group Regular Point 
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Literature Cited

  1. 1.
    A. A. Agrachev, R. V. Gamkreilidze, and A. V. Sarychev “Local invariants of smooth control systems,”Acta Appl. Math.,14, 191–237 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Bellaïche, J.-P. Laumond, and J.-J. Risler, “Nilpotent infinitesimal approximations to a control Lie algebra,” In:IFAC Nonlinear Control Systems Design Symposium, Bordeaux, France, June (1992), pp. 174–181.Google Scholar
  3. 3.
    A. Connes, “Noncommutative geometry,” Preprint (1993).Google Scholar
  4. 4.
    N. Goodman, “Nilpotent Lie groups,”Lect. Notes Math.,562 (1976).Google Scholar
  5. 5.
    M. Gromov,Structures Métriques pour les Variétés Riemanniennes, Cedic-Nathan, Paris (1981).zbMATHGoogle Scholar
  6. 6.
    M. Gromov, “Carnot-Carathéodory spaces seen from within,” Preprint (1994).Google Scholar
  7. 7.
    V. V. Grushin, “On a class of hypoelliptic operators”,Mat. Sb.,12, 458–476 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    V. V. Grushin, “A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,”Mat. Sb.,13, 155–185 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    B. Helffer and J. Nourrigat, “Approximation d’un système de champs de vecteurs, et applications à l’hypoellipticité”,Ark. Mat.,19, 237–254 (1979).CrossRefzbMATHGoogle Scholar
  10. 10.
    R. Hermann, “Geodesics of singular Riemannian metrics,”Bull. Am. Math. Soc.,79, 780–782 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    H. Hermes, “Nilpotent and high-order approximations of vector field systems”,SIAM Rev.,33, No. 2, 238–264 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J. Mitchell, “On Carnot-Carathéodory metrics,”J. Diff. Geom. 21, 35–45 (1985).CrossRefzbMATHGoogle Scholar
  13. 13.
    P. Pansu, “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,”Ann. Math.,129, No. 2, 1–60 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ch. Rockland, “Intrinsic nilpotent approximations,”Acta Appl. Math.,8, 213–270 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    L. P. Rothschild and E. M. Stein, “Hypoelliptic differential operators and nilpotent groups,”Acta Math.,137, 247–320 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    E. M. Stein, “Accessible sets, orbits, and foliations with singularities,”Proc. London Math. Soc.,29, 699–713 (1974).MathSciNetzbMATHGoogle Scholar
  17. 17.
    E. M. Stein, “Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups,” In:Actes du Congrès International des Mathématiciens, Vol. 1. Nice (1970), pp. 173–189.Google Scholar
  18. 18.
    E. M. Stein,Harmonic Analysis Princeton University Press (1993).Google Scholar
  19. 19.
    R. S. Strichartz, “Sub-Riemannian geometry.”J. Diff. Geom.,24, 221–263 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    H. J. Sussmann and V. Jurdjevic, “Controllability of nonlinear systems,”J. Diff. Equat.,12, 95–116 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    H. J. Sussmann, “Orbits of families of vector fields and integrability of distributions,”Trans. Am. Math. Soc.,180, 171–188 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A. N. Varchenko, “Obstruction to local equivalence of distributions,”Mat. Zametki,29, 479–484 (1981)MathSciNetzbMATHGoogle Scholar
  23. 23.
    N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon,Analysis and Geometry on Groups, Cambridge University Press (1992).Google Scholar
  24. 24.
    A. M. Vershik and V. Ya. Gershkovich, “Nonholomic problems and the theory of distributions,”Acta Appl. Math.,12, 181–209 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. M. Vershik and V. Ya. Gershkovich, “Nonholonomic dynamical systems, geometry of distributions and variational problems,” In:Dynamical Systems. VII,Encyclopaedia of Mathematical Sciences. Vol. 16, Springer (in press).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

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  • A. Bellaïche

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