Coordinates Adapted to Vector Fields III: Real Analyticity

TL;DR

This paper provides necessary and sufficient conditions for when a set of vector fields on a manifold can be transformed into real analytic coordinates, with a detailed quantitative analysis, advancing the understanding of coordinate systems in sub-Riemannian geometry.

Contribution

It establishes coordinate-free criteria for real analyticity of vector fields and offers a quantitative study of the associated coordinate charts, extending previous work on smooth and Zygmund spaces.

Findings

Necessary and sufficient conditions for real analytic coordinate systems.

Quantitative analysis of coordinate charts.

Extension of previous smooth and Zygmund space 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 real analytic. 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 third part in a three-part series of papers. The first part, joint with Stovall, lay the groundwork for the coordinate system we use in this paper and showed how such coordinate charts can be viewed as scaling maps for sub-Riemannian geometry. The second part dealt with the analogous questions with real analytic replaced by $C^\infty$ and Zygmund spaces.