Lipschitz classification of almost-Riemannian distances on compact oriented surfaces

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--Caratheodory 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 analyse 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 labelled graph associated with it.
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Submitted on : Wednesday, March 17, 2010 - 10:22:03 AM
Last modification on : Saturday, March 30, 2019 - 1:18:10 AM
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  • HAL Id : hal-00464414, version 1
  • ARXIV : 1003.4842

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Ugo Boscain, Grégoire Charlot, Roberta Ghezzi, Mario Sigalotti. Lipschitz classification of almost-Riemannian distances on compact oriented surfaces. Journal of Geometric Analysis, 2013, 23, pp. 438-455. ⟨hal-00464414⟩

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