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

Abstract

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 \(G\).

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Notes

  1. 1.

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

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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 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.

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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

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Keywords

  • Carnot manifolds
  • Privileged coordinates
  • Nilpotent approximation

Mathematics Subject Classification (2010)

  • 53C17
  • 43A85
  • 22E25