Duke Mathematical Journal

The Sard conjecture on Martinet surfaces

André Belotto da Silva and Ludovic Rifford

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Abstract

Given a totally nonholonomic distribution of rank 2 on a 3-dimensional manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from the same point. In this setting, by the Sard conjecture, that set should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces, and we show that the result holds true under an assumption of nontransversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.

Article information

Source
Duke Math. J., Volume 167, Number 8 (2018), 1433-1471.

Dates
Received: 20 August 2016
Revised: 6 October 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1520046164

Digital Object Identifier
doi:10.1215/00127094-2017-0058

Mathematical Reviews number (MathSciNet)
MR3807314

Zentralblatt MATH identifier
06896950

Subjects
Primary: 53A99: None of the above, but in this section
Secondary: 34H05: Control problems [See also 49J15, 49K15, 93C15] 32S45: Modifications; resolution of singularities [See also 14E15]

Keywords
Sard conjecture sub-Riemannian geometry differential geometry differential forms control theory resolution of singularities

Citation

Belotto da Silva, André; Rifford, Ludovic. The Sard conjecture on Martinet surfaces. Duke Math. J. 167 (2018), no. 8, 1433--1471. doi:10.1215/00127094-2017-0058. https://projecteuclid.org/euclid.dmj/1520046164


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