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Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces

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Abstract

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot–Carathéodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyze the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.

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Authors and Affiliations

  1. CNRS CMAP & Team INRIA GECO, École Polytechnique Route de Saclay, 91128, Palaiseau Cedex, France

    U. Boscain

  2. Institut Fourier & Team INRIA GECO, UMR 5582, CNRS/Université Grenoble 1, 100 rue des Maths, BP 74, 38402, St Martin d’Hères, France

    G. Charlot

  3. CMAP & Team INRIA GECO, École Polytechnique Route de Saclay, 91128, Palaiseau Cedex, France

    R. Ghezzi

  4. Team INRIA GECO, INRIA Saclay–Île-de-France, Parc Orsay Université, 4 rue Jacques Monod, 91893, Orsay Cedex, France

    M. Sigalotti

  5. CMAP, École Polytechnique Route de Saclay, 91128, Palaiseau Cedex, France

    M. Sigalotti

Authors
  1. U. Boscain
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  2. G. Charlot
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  3. R. Ghezzi
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  4. M. Sigalotti
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Corresponding author

Correspondence to R. Ghezzi.

Additional information

Communicated by Jiaping Wang.

This research has been supported by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number 239748, and by the ANR “GCM”, program “Blanc–CSD” project number NT09-504490.

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Boscain, U., Charlot, G., Ghezzi, R. et al. Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces. J Geom Anal 23, 438–455 (2013). https://doi.org/10.1007/s12220-011-9262-4

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  • DOI: https://doi.org/10.1007/s12220-011-9262-4

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