Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinates
Abstract
This paper is a sequel of Choi and Ponge (J Dyn Control Syst 25:109–157, 2019) and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold, it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single out a special class of privileged coordinates in which the nilpotent approximation at a given point of a Carnot manifold is given by its tangent group. We call these coordinates Carnot coordinates. Examples of Carnot coordinates include Darboux coordinates on contact manifolds and the canonical coordinates of the first kind of Goodman and Rothschild-Stein. By converting the privileged coordinate of Bellaïche into Carnot coordinates, we obtain an effective construction of Carnot coordinates, which we call ε-Carnot coordinates. They form the building block of all systems of Carnot coordinates. On a graded nilpotent Lie group, they are given by the group law of the group. For general Carnot manifolds, they depend smoothly on the base point. Moreover, in Carnot coordinates at a given point, they are osculated in a very precise manner by the group law of the tangent group at the point.
References
[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]
Agrachev A, Gamkrelidze R, Sarychev A. Local invariants of smooth control systems. Acta Appl Math 1989;14:191–237.
[3]
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).
[4]
Beals R, Greiner P, Vol. 119. Calculus on Heisenberg manifolds annals of mathematics studies. Princeton: Princeton University Press; 1988.
[5]
Bellaïche A, Vol. 144. The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry. Progr Math. Birkhäuser: Basel; 1996, pp. 1–78.
[6]
Bianchini RM, Stefani C. Graded approximations and controllability along a trajectory. SIAM J Control Optim 1990;28:903–924.
[7]
Choi W, Ponge R. Privileged coordinates and nilpotent approximation for Carnot manifolds, I. General results. J Dyn Control Syst 2019;25:109–157.
[8]
Choi W, Ponge R. Tangent maps and tangent groupoid for Carnot manifolds. Differential Geom Appl. 2019;62:136–183.
[9]
Choi W, Ponge R. A Pseudodifferential calculus on Carnot manifolds. In preparation.
[10]
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.
[11]
Goodman N, Vol. 562. Nilpotent Lie groups: structure and applications to analysis. Lecture notes in mathematics. Berlin: Springer; 1976.
[12]
Gromov M, Vol. 144. Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry. Progr. Math. Birkhäuser: Basel; 1996, pp. 85–323.
[13]
Jean F. Control of nonholonomic systems: from sub-Riemannian geometry to motion planning. Springer briefs in mathematics. New York: Springer International Publishing; 2014.
[14]
Hermes H. Nilpotent and high-order approximations of vector field systems. SIAM Rev 1991;33:238–264.
[15]
Margulis G, Mostow GD. Some remarks on the definition of tangent cones in a Carnot-Carathéodory space. J Anal Math 2000;80:299–317.
[16]
Métivier G. Function spectrale et valeurs propres d’une classe d’opérateurs non elliptiques. Comm Partial Differ Equ 1976;47:467–510.
[17]
Mitchell J. On Carnot-Carathéodory metrics. J Differential Geom 1985;21:35–45.
[18]
Montgomery R, Vol. 91. A tour of subriemannian geometries, their geodesics and applications. Mathematical surveys and monographs. Providence: American Mathematical Society; 2002.
[19]
Nagel A, Stein E, Wainger S. Balls and metrics defined by vector fields. I. Basic properties. Acta Math 1985;155:103–147.
[20]
Ponge R. The tangent groupoid of a Heisenberg manifold. Pacific J Math 2006; 227(1):151–175.
[21]
Rothschild L, Stein E. Hypoelliptic differential operators and nilpotent groups. Acta Math 1976;137(3-4):247–320.
[22]
Stefani G. On the local controllability of a scalar-input control system. Theory and applications of nonlinear control systems (Stockholm, 1985). North-Holland. 1986;167–179.
[23]
Tanaka N. On differential systems, graded Lie algebras and pseudogroups. J Math Kyoto Univ 1970;10:1–82.
Recommendations
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 ...
Rigid Carnot Algebras: A Classification
A Carnot algebra is a graded nilpotent Lie algebra L = L 1 L r generated by L 1 . The bidimension of the Carnot algebra L is the pair (dim L 1 , dim L ). A Carnot algebra is said to be rigid if it is isomorphic to any of its small ...
Classification of Carnot Algebras: the Semi-Rigid Cases
A Carnot algebra is a graded nilpotent Lie algebra L = L 1 L r generated by L 1 . The bidimension of the Carnot algebra L is the pair (dim L 1 , dim L ). A Carnot algebra is said to be rigid if it is isomorphic to any of its small ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
Copyright © 2019 Springer Science+Business Media, LLC, part of Springer Nature.
Publisher
Kluwer Academic Publishers
United States
Publication History
Published: 01 October 2019
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 12 Jan 2025