Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results
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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 \(G\).
KeywordsCarnot manifolds Privileged coordinates Nilpotent approximation
Mathematics Subject Classification (2010)53C17 43A85 22E25
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.
- 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.
- 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.)Google Scholar
- 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.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.
- 8.Beals R, Greiner P. Calculus on Heisenberg manifolds. Annals of mathematics studies, Vol. 119. Princeton: Princeton University Press; 1988.Google Scholar
- 9.Bellac̈he A. The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry, pp. 1–78, Progr. Math., Vol. 144. Birkhäuser: Basel; 1996.Google Scholar
- 11.Biquard O. 1999. Quaternionic contact structures. Quaternionic structures in mathematics and physics, Univ. Studi Roma La Sapienza, Rome.Google Scholar
- 12.Biquard O. 2000. Métriques d’Einstein asymptotiquement symétriques, Vol. 265 of Astérisque, p. 115.Google Scholar
- 13.Bloch AM. Nonholonomic mechanics and control. Interdisciplinary applied mathematics, Vol. 24. New York: Springer; 2003.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Choi W, Ponge R. Tangent maps and tangent groupoid for Carnot manifolds. arXiv:1510.05851v2 (v2: September 2017), p. 40.
- 22.Choi W, Ponge R. A pseudodifferential calculus on Carnot manifolds. In preparation.Google Scholar
- 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.Google Scholar
- 27.Eliashberg Y, Thurston W. Confoliations. University lecture series, Vol. 13. Providence: AMS; 1998.Google Scholar
- 29.Fischer V, Ruzhansky M. 2016. Quantization on nilpotent Lie groups. Progress in Mathematics, 314. Birkhäuser/Springer.Google Scholar
- 34.Fox DJF. Contact path geometries. arXiv:math.DG/0508343, p. 36.
- 37.Gromov M. Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, pp. 85–323, Progr. Math., Vol. 144. Birkhäuser: Basel; 1996.Google Scholar
- 41.Jean F. Control of nonholonomic systems: from sub-Riemannian geometry to motion planning. Springer briefs in mathematics. New York: Springer International Publishing; 2014.Google Scholar
- 44.Melin A. 1982. Lie filtrations and pseudo-differential operators. Preprint.Google Scholar
- 49.Montgomery R. A tour of subriemannian geometries, their geodesics and applications. Mathematical surveys and monographs, Vol. 91. Providence: American Mathematical Society; 2002.Google Scholar
- 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.Google Scholar
- 59.van Erp E. Contact structures of arbitrary codimension and idempotents in the Heisenberg algebra. arXiv:1001.5426, p. 13.