Coordinates Adapted to Vector Fields II: Sharp Results

TL;DR

This paper establishes precise, coordinate-free conditions for when a set of vector fields on a manifold can be smoothly transformed into a coordinate system with optimal regularity, advancing the understanding of sub-Riemannian geometry.

Contribution

It provides necessary and sufficient conditions for the existence of highly regular coordinate charts adapted to vector fields, with a sharp, quantitative analysis, extending previous sub-Riemannian results.

Findings

Necessary and sufficient conditions for coordinate regularity

Quantitative analysis of coordinate charts

Generalization of sub-Riemannian geometry results

Abstract

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are $\mathscr{C}^{s+1}$ for $s\in (1,\infty]$, where $\mathscr{C}^s$ denotes the Zygmund space of order $s$. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the second part in a three-part series of papers. The first part, joint with Stovall, addressed the same question, though the results were not sharp, and showed how such coordinate charts can be viewed as scaling maps in sub-Riemannian geometry. When viewed in this light, these results can be seen as strengthening and generalizing previous works on the quantitative theory of sub-Riemannian geometry, initiated by Nagel, Stein, and Wainger, and furthered by Tao and Wright, the author, and others. In the third part, we prove similar results concerning real analyticity.