Differentiability of mappings in the geometry of Carnot manifolds

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.”