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Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces

TitleLipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces
Publication TypeJournal Article
Year of Publication2013
AuthorsBoscain, U, Charlot, G, Ghezzi, R, Sigalotti, M
JournalJournal of Geometric Analysis
Volume23
Pagination438–455
Date PublishedJan
ISSN1559-002X
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.

URLhttps://doi.org/10.1007/s12220-011-9262-4
DOI10.1007/s12220-011-9262-4

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