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Coordinates Adapted to Vector Fields: Canonical Coordinates

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Abstract

Given a finite collection of C1 vector fields on aC2 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–Carathéodory) 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 form 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.

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Acknowledgments

We thank the referee whose detailed comments improved the exposition. Stovall was partially supported by National Science Foundation Grant No. 1600458. Street was partially supported by National Science Foundation Grant Nos. 1401671 and 1764265. This material is partially based upon work supported by the National Science Foundation under Grant No. 1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the spring semester of 2017.

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  1. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI, 53706, USA

    Betsy Stovall & Brian Street

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  1. Betsy Stovall
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  2. Brian Street
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Correspondence to Brian Street.

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Stovall, B., Street, B. Coordinates Adapted to Vector Fields: Canonical Coordinates. Geom. Funct. Anal. 28, 1780–1862 (2018). https://doi.org/10.1007/s00039-018-0469-4

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