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

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We study the differentiability of mappings in the geometry of Carnot-Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of hc-differentiability and prove the hc-differentiability of Lipschitz mappings of Carnot-Carathéodory spaces (a generalization of Rademacher's theorem) and a generalization of Stepanov's theorem. As a consequence, we obtain the hc-differentiability almost everywhere of the quasiconformal mappings of Carnot-Carathéodory spaces. We establish the hc-differentiability of rectifiable curves by way of proof. Moreover, the paper contains a new proof of the functorial property of the correspondence "a local basis → the nilpotent tangent cone."