Geometry of Carnot--Carath\'eodory Spaces, Differentiability and Coarea Formula
Abstract
We give a simple proof of Gromov's Theorem on nilpotentization of vector fields, and exhibit a new method for obtaining quantitative estimates of comparing geometries of two different local Carnot groups in Carnot--Carath\'eodory spaces with C1,α-smooth basis vector fields, α∈[0,1]. From here we obtain the similar estimates for comparing geometries of a Carnot--Carath\'eodory space and a local Carnot group. These two theorems imply basic results of the theory: Gromov type Local Approximation Theorems, and for α>0 Rashevski-Chow Theorem and Ball--Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot--Carath\'eodory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for some classes of contact mappings of Carnot--Carath\'eodory spaces.