Coordinates adapted to vector fields III: Real analyticity

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.

Keywords

vector fields, real analytic, sub-Riemannian, Carnot–Carathéodory, scaling

2010 Mathematics Subject Classification

Primary 58A30. Secondary 32C05, 53C17.

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This material is partially based upon work supported by the National Science Foundation under Grant No. 1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the spring semester of 2017. The author was also partially supported by National Science Foundation Grant Nos. 1401671 and 1764265.

Received 19 February 2020

Accepted 5 March 2020

Published 3 September 2021