, Volume 21, Issue 6, pp 1358-1374

A New Approach to Investigation of Carnot–Carathéodory Geometry

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Abstract

We develop a new approach to studying the geometry of Carnot–Carathéodory spaces under minimal assumptions on the smoothness of basis vector fields. We obtain quantitative comparison estimates for the local geometries of two different local Carnot groups, as well as of a local Carnot group and the original space. As corollaries, we deduce some results that are well-known and basic for the “smooth” case: the generalized triangle inequality for d , the local approximation theorem for the quasimetric d , the Rashevskiǐ–Chow theorem, the ball-box theorem, and so on.