Privileged coordinates for Carnot-Carath\'eodory spaces of low smoothness

TL;DR

This paper introduces coordinate systems in low-smoothness Carnot-Carathéodory spaces that enable homogeneous approximations, establishing minimal smoothness conditions and extending results from smooth cases.

Contribution

It defines new coordinate classes for low-smoothness spaces, identifies minimal smoothness for their equivalence to privileged coordinates, and applies these to canonical coordinate results.

Findings

Established minimal smoothness thresholds for coordinate class equivalence

Developed convergence theorems in quasimetric spaces

Extended privileged coordinate concepts to low-smoothness contexts

Abstract

We describe classes of coordinate systems in Carnot-Carath\'eodory spaces of low smoothness which allow for homogeneous approximations of quasimetrics and basis vector fields. We establish the minimal smoothness required for these classes to coinside with the class of the privileged coordinates described earlier for the smooth case. We also apply these results to prove partial analogues of existing results in the canonical coordinates of the 2nd kind. As a geometric tool we prove some convergence theorems in quasimetric spaces.