Geometry of Carnot-Carathéodory Spaces, Differentiability, Coarea and Area Formulas

Abstract

We compare geometries of two different local Lie groups in a Carnot-Carathéodory space, and obtain quantitative estimates of their difference. This result is extended to Carnot-Carathéodory spaces with C1,α-smooth basis vector fields, α ∈ [0, 1], and the dependence of the estimates on α is established. From here we obtain the similar estimates for comparing geometries of a Carnot-Carathéodory space and a local Lie group. These results base on Gromov’s Theorem on nilpotentization of vector fields for which we give new and simple proof. All the above 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 smooth contact mappings of Carnot-Carathéodory spaces, and the area formula for Lipschitz (with respect to sub-Riemannian metrics) mappings of Carnot-Carathéodory spaces.