Coordinates Adapted to Vector Fields: Canonical Coordinates
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 subRiemannian (aka CarnotCarathéodory) 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1709.04528
 Bibcode:
 2017arXiv170904528S
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Metric Geometry;
 58A30 (Primary);
 57R55 and 53C17 (Secondary)
 EPrint:
 Part 1 in a 3 part series, 64 pages. final version, to appear in GAFA