Coordinates Adapted to Vector Fields: Canonical Coordinates

TL;DR

This paper investigates conditions under which a coordinate system can be found to enhance the smoothness of a collection of vector fields on a manifold, providing a foundational step for a series of studies on sub-Riemannian geometry.

Contribution

It introduces a coordinate system adapted to vector fields that allows for a quantitative analysis of their smoothness properties, extending previous sub-Riemannian geometry theories.

Findings

Provides a non-sharp, foundational result on adapted coordinate systems.

Establishes a diffeomorphism invariant framework for the theory.

Lays groundwork for sharp results in subsequent papers.

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 have a higher level of smoothness. For example, when is there a coordinate system in which the vector fields are smooth, or real analytic, or have Zygmund regularity of some finite order? We address this question in a quantitative way, which strengthens and generalizes previous works on the quantitative theory of sub-Riemannian (aka Carnot-Carath\'eodory) geometry due to Nagel, Stein, and Wainger, Tao and Wright, the second author, and others. Furthermore, we provide a diffeomorphism invariant version of these theories. This is the first part in a three part series of papers. In this paper, we study a particular coordinate system adapted to a collection of vector fields (sometimes called canonical coordinates) and present results related to the above questions which are not quite sharp; these results from the backbone of the series. The methods of this paper are based on techniques from ODEs. In the second paper, we use additional methods from PDEs to obtain the sharp results. In the third paper, we prove results concerning real analyticity and use methods from ODEs.