紧定向曲面上几乎黎曼距离的Lipschitz分类
Lipschitz classification of almost-Riemannian distances on compact oriented surfaces
摘要：
二维准黎曼结构是曲面上的广义黎曼结构，其局部正交标架由可以共线的向量场的李括号生成对给出。我们认为卡诺-Caratheodory距离规范与几乎黎曼结构和研究的Lipschitz等价问题的两个这样的距离在同一个紧凑的定向表面。我们分析了一般情况，特别是允许切点的存在，即，点的两个发电机的分布和他们的李括号是线性相关的。本文的主要结果提供了几乎黎曼距离的Lipschitz等价类关于与之相关的标号图的一个刻画。
英文摘要：
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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