Geometric and Functional Analysis

, Volume 21, Issue 6, pp 1358–1374

A New Approach to Investigation of Carnot–Carathéodory Geometry


DOI: 10.1007/s00039-011-0142-7

Cite this article as:
Karmanova, M. Geom. Funct. Anal. (2011) 21: 1358. doi:10.1007/s00039-011-0142-7


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.

Keywords and phrases

Carnot manifold local Carnot group local geometry minimal smoothness 

2010 Mathematics Subject Classification

Primary 53C17 51F99 

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics of SBNovosibirskRussia