Geometry of Carnot--Carath\'{e}odory Spaces, Differentiability and Coarea Formula

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\'{e}odory spaces with $C^{1,\alpha}$-smooth basis vector fields, $\alpha\in[0,1]$. From here we obtain the similar estimates for comparing geometries of a Carnot--Carath\'{e}odory space and a local Carnot group. These two theorems imply basic results of the theory: Gromov type Local Approximation Theorems, and for $\alpha>0$ Rashevski\v{\i}-Chow Theorem and Ball--Box Theorem, etc. We apply the obtained results for proving $hc$-differentiability of mappings of Carnot--Carath\'{e}odory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for some classes of contact mappings of Carnot--Carath\'{e}odory spaces.