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 $\mapsto$ the nilpotent tangent cone."