Lipschitz classification of almost-Riemannian distances on compact oriented surfaces

TL;DR

This paper characterizes when two almost-Riemannian distances on compact surfaces are Lipschitz equivalent by using a graph-based classification, even in the presence of tangency points.

Contribution

It introduces a graph-based method to classify Lipschitz equivalence classes of almost-Riemannian distances on surfaces, including cases with tangency points.

Findings

Lipschitz equivalence characterized by associated labelled graphs

Includes analysis of tangency points in the classification

Provides a complete classification framework for almost-Riemannian distances

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.