Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results


In this paper, we attempt to give a systematic account on privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold, it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel (Choi and Ponge 2017) by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group \(G\) satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by \(G\).

This is a preview of subscription content, log in to check access.


  1. 1.

    The terminology “pluri-contact manifold” is not used in [5]. We use it for the sake of exposition’s clarity.


  1. 1.

    Agrachev A, Barilari D, Boscain U. Introduction to Riemannian and sub-Riemannian geometry. To appear, http://webusers.imj-prg.fr/~davide.barilari/Notes.php.

  2. 2.

    Agrachev A, Marigo A. Nonholonomic tangent spaces: intrinsic construction and rigid dimensions. Electron Res Announc Amer Math Soc 2003;9:111–120.

  3. 3.

    Agrachev A, Marigo A. Rigid Carnot algebras: a classification. J Dyn Control Syst 2005;11:449–494.

  4. 4.

    Agrachev AA, Sarychev AV. Filtrations of a Lie algebra of vector fields and nilpotent approximations of control systems. Dokl Akad Nauk SSSR 1987;285:777–781. (English transl.: Soviet Math. Dokl. 36 (1988), 104–108.)

  5. 5.

    Apostolov V, Calderbank DMJ, Gauduchon P, Legendre E. Toric contact geometry in arbitrary codimension. arXiv:1708.04942, p. 22. To appear in Int Math Res Notices.

  6. 6.

    Apostolov V, Calderbank DMJ, Gauduchon P, Legendre E. Levi-Kähler reduction of CR structures, products of spheres, and toric geometry. arXiv:1708.05253, p. 39.

  7. 7.

    Banyaga A. On essential conformal groups and a conformal invariant. J Geom 2000; 68:10–15.

  8. 8.

    Beals R, Greiner P. Calculus on Heisenberg manifolds. Annals of mathematics studies, Vol. 119. Princeton: Princeton University Press; 1988.

  9. 9.

    Bellac̈he A. The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry, pp. 1–78, Progr. Math., Vol. 144. Birkhäuser: Basel; 1996.

  10. 10.

    Bianchini RM, Stefani C. Graded approximations and controllability along a trajectory. SIAM J Control Optim 1990;28:903–924.

  11. 11.

    Biquard O. 1999. Quaternionic contact structures. Quaternionic structures in mathematics and physics, Univ. Studi Roma La Sapienza, Rome.

  12. 12.

    Biquard O. 2000. Métriques d’Einstein asymptotiquement symétriques, Vol. 265 of Astérisque, p. 115.

  13. 13.

    Bloch AM. Nonholonomic mechanics and control. Interdisciplinary applied mathematics, Vol. 24. New York: Springer; 2003.

  14. 14.

    Bonfiglioli A, Lanconelli E, Uguzzoni F. Stratified Lie groups and potential theory for their sub-Laplacians. Springer monographs in mathematics. Berlin: Springer; 2007.

  15. 15.

    Bramanti M. An invitation to hypoelliptic operators and Hörmander’s vector fields. SpringerBriefs in mathematics. New York: Springer International Publishing; 2014.

  16. 16.

    Bryant R. Conformal geometry and 3-plane fields on 6-manifolds. Developments of Cartan geometry and related mathematical problems. RIMS symposium proceedings, Vol. 1502, pp. 1–15; 2006.

  17. 17.

    Calin O, Chang D.-C. Sub-riemannian geometry. General theory and examples encyclopedia of mathematics and its applications, Vol. 126. Cambridge: Cambridge University Press; 2009.

  18. 18.

    Cartan E. Les systèmes de Pfaff à cinq variables et les équations aux derivées partielles du second ordre. Ann Sci École Norm Sup 1910;27:263–355.

  19. 19.

    Čap A, Slovák J. Parabolic geometries I: background and general theory. Mathematical surveys and mo- nographs, Vol. 154. Providence: American Mathematical Society; 2009. p. 628, ISBN: 0-8218-2681-6.

  20. 20.

    Choi W, Ponge R. Privileged coordinates and nilpotent approximation for Carnot manifolds, II. Carnot coordinates. arXiv:1703.05494v2 (v2: September 2017), p. 36.

  21. 21.

    Choi W, Ponge R. Tangent maps and tangent groupoid for Carnot manifolds. arXiv:1510.05851v2 (v2: September 2017), p. 40.

  22. 22.

    Choi W, Ponge R. A pseudodifferential calculus on Carnot manifolds. In preparation.

  23. 23.

    Connes A. Noncommutative geometry. San Diego: Academic; 1994.

  24. 24.

    Connes A, Moscovici H. The local index formula in noncommutative geometry. Geom Funct Anal 1995;5:174–243.

  25. 25.

    Corwin L, Greenleaf F. Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples. Cambridge studies in advanced mathematics, Vol. 18. Cambridge: Cambridge University Press; 1990.

  26. 26.

    Cummins T. A pseudodifferential calculus associated with 3-step nilpotent groups. Comm Partial Differ Equ 1989;14(1):129–171.

  27. 27.

    Eliashberg Y, Thurston W. Confoliations. University lecture series, Vol. 13. Providence: AMS; 1998.

  28. 28.

    Falbel E, Jean F. Measures of transverse paths in sub-Riemannian geometry. J Anal Math 2003;91:231–246.

  29. 29.

    Fischer V, Ruzhansky M. 2016. Quantization on nilpotent Lie groups. Progress in Mathematics, 314. Birkhäuser/Springer.

  30. 30.

    Folland GB. Lipschitz classes and Poisson integrals on stratified groups. Stud Math 1979;66:37–55.

  31. 31.

    Folland G, Stein E. Estimates for the \(\overline {\partial }_{b}\)-complex and analysis on the Heisenberg group. Comm Pure Appl Math 1974;27:429–522.

  32. 32.

    Folland G, Stein E. Hardy spaces on homogeneous groups. Mathematical notes, 28. Princeton: Princeton University Press; 1982.

  33. 33.

    Fox DJF. Contact projective structures. Indiana Univ Math J 2005;54:1547–1598.

  34. 34.

    Fox DJF. Contact path geometries. arXiv:math.DG/0508343, p. 36.

  35. 35.

    Gershkovich V, Vershik A. Nonholonomic manifolds and nilpotent analysis. J Geom Phys 1988;5:407–452.

  36. 36.

    Goodman N. Nilpotent Lie groups: structure and applications to analysis. Lecture notes in mathematics, Vol. 562. Berlin: Springer; 1976.

  37. 37.

    Gromov M. Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, pp. 85–323, Progr. Math., Vol. 144. Birkhäuser: Basel; 1996.

  38. 38.

    Hermes H. Nilpotent and high-order approximations of vector field systems. SIAM Rev 1991;33:238–264.

  39. 39.

    Hörmander L. Hypoelliptic second order differential equations. Acta Math 1967; 119:147–171.

  40. 40.

    Jean F. The car with N trailers: characterization of the singular configurations. ESAIM: Cont Opt Calc Var 1996;1: 241–266.

  41. 41.

    Jean F. Control of nonholonomic systems: from sub-Riemannian geometry to motion planning. Springer briefs in mathematics. New York: Springer International Publishing; 2014.

  42. 42.

    Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans Am Math Soc 1980;258:147–153.

  43. 43.

    Margulis G, Mostow GD. Some remarks on the definition of tangent cones in a Carnot-Carathéodory space. J Anal Math 2000;80:299–317.

  44. 44.

    Melin A. 1982. Lie filtrations and pseudo-differential operators. Preprint.

  45. 45.

    Métivier G. Function spectrale et valeurs propres d’une classe d’opérateurs non elliptiques. Comm Partial Differ Equ 1976;47:467–510.

  46. 46.

    Métivier G. Hypoellipticité analytique sur des groupes nilpotents de rang 2. Duke Math J 1980;47:195–213.

  47. 47.

    Mitchell J. On Carnot-Carathéodory metrics. J Differ Geom 1985;21:35–45.

  48. 48.

    Moerdijk I, Mrčun J. Introduction to foliations and Lie groupoids. Cambridge studies in advanced mathematics, Vol. 91. Cambridge: Cambridge University Press; 2003.

  49. 49.

    Montgomery R. A tour of subriemannian geometries, their geodesics and applications. Mathematical surveys and monographs, Vol. 91. Providence: American Mathematical Society; 2002.

  50. 50.

    Nagel A, Stein EM, Wainger S. Metrics defined by vector fields I: basic properties. Acta Math 1985;155:103–147.

  51. 51.

    Pansu P. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann Math (2) 1989;129:1–60.

  52. 52.

    Perko L. Differential equations and dynamical systems. New York: Springer; 2001.

  53. 53.

    Ponge R. The tangent groupoid of a Heisenberg manifold. Pacific J Math 2006; 227(1):151–175.

  54. 54.

    Rifford L. Sub-Riemannian geometry and optimal transport. Springer briefs in mathematics. New York: Springer International Publishing; 2014.

  55. 55.

    Rockland C. Intrinsic nilpotent approximation. Acta Appl Math 1987;8(3):213–270.

  56. 56.

    Rothschild L, Stein E. Hypoelliptic differential operators and nilpotent groups. Acta Math 1976;137(3-4):247–320.

  57. 57.

    Stefani G. On local controllability of the scalar input control systems. Analysis and control of nonlinear systems. Amsterdam: North-Holland; 1988, pp. 213–220.

  58. 58.

    Tanaka N. On differential systems, graded Lie algebras and pseudogroups. J Math Kyoto Univ 1970;10:1–82.

  59. 59.

    van Erp E. Contact structures of arbitrary codimension and idempotents in the Heisenberg algebra. arXiv:1001.5426, p. 13.

  60. 60.

    Vergne M. Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes. Bull Soc Math France. 1970;98:81–116.

  61. 61.

    Vershik A, Gershkovich V. A bundle of nilpotent Lie algebras over a nonholonomic manifold. J Soviet Math 1992;59:1040–1053.

  62. 62.

    Weinstein A. Fat bundles and symplectic manifolds. Adv Math 1980;37:239–250.

Download references


The authors wish to thank Andrei Agrachev, Davide Barilari, Enrico Le Donne, and Frédéric Jean for useful discussions related to the subject matter of this paper. They also thank an anonymous referee whose insightful comments help improving the presentation of the paper. In addition, they would like to thank Henri Poincaré Institute (Paris, France), McGill University (Montréal, Canada) and University of California at Berkeley (Berkeley, USA) for their hospitality during the preparation of this paper.

Author information

Correspondence to Raphaël Ponge.

Additional information

WC was partially supported by POSCO TJ Park Foundation. RP was partially supported by Research Resettlement Fund and Foreign Faculty Research Fund of Seoul National University, and Basic Research grants 2013R1A1A2008802 and 2016R1D1A1B01015971 of the National Research Foundation of Korea.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Choi, W., Ponge, R. Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results. J Dyn Control Syst 25, 109–157 (2019). https://doi.org/10.1007/s10883-018-9404-0

Download citation


  • Carnot manifolds
  • Privileged coordinates
  • Nilpotent approximation

Mathematics Subject Classification (2010)

  • 53C17
  • 43A85
  • 22E25