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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.

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Acknowledgements

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 anonymous referees 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.

Funding

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

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Correspondence to Raphaël Ponge.

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Choi, W., Ponge, R. Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinates. J Dyn Control Syst 25, 631–670 (2019). https://doi.org/10.1007/s10883-019-09434-9

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Keywords

  • Carnot manifolds
  • Privileged coordinates
  • Nilpotent approximation

Mathematics Subject Classification (2010)

  • 53C17
  • 43A85
  • 22E25