Coordinates Adapted to Vector Fields III: Real Analyticity
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, coordinatefree 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 threepart 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 subRiemannian geometry. The second part dealt with the analogous questions with real analytic replaced by $C^\infty$ and Zygmund spaces.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 arXiv:
 arXiv:1808.04635
 Bibcode:
 2018arXiv180804635S
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Classical Analysis and ODEs;
 2010: 58A30 (Primary);
 32C05 and 53C17 (Secondary)
 EPrint:
 44 pages