Approximate Differentiability of Mappings of Carnot-Carath\'eodory Spaces

TL;DR

This paper investigates the approximate differentiability of measurable mappings in Carnot--Carathéodory spaces, establishing equivalence with differentiability along horizontal vector fields and generalizing classical theorems to these geometric contexts.

Contribution

It extends classical Euclidean and Carnot group differentiability theorems to Carnot--Carathéodory spaces, including a generalized Rashevsky--Chow theorem.

Findings

Approximate differentiability almost everywhere is equivalent to differentiability along horizontal vector fields.

Generalization of Rashevsky--Chow theorem for $C^1$-smooth vector fields.

Extension of Stepanoff, Whitney, and Vodopyanov theorems to Carnot--Carathéodory spaces.

Abstract

We study the approximate differentiability of measurable mappings of Carnot--Carath\'eodory spaces. We show that the approximate differentiability almost everywhere is equivalent to the approximate differentiability along the basic horizontal vector fields almost everywhere. As a geometric tool we prove the generalization of Rashevsky--Chow theorem for $C^1$-smooth vector fields. The main result of the paper extends theorems on approximate differentiability proved by Stepanoff (1923, 1925) and Whitney (1951) in Euclidean spaces and by Vodopyanov (2000) on Carnot groups.