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1 June 2018 The Sard conjecture on Martinet surfaces
André Belotto da Silva, Ludovic Rifford
Duke Math. J. 167(8): 1433-1471 (1 June 2018). DOI: 10.1215/00127094-2017-0058

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.

Citation

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André Belotto da Silva. Ludovic Rifford. "The Sard conjecture on Martinet surfaces." Duke Math. J. 167 (8) 1433 - 1471, 1 June 2018. https://doi.org/10.1215/00127094-2017-0058

Information

Received: 20 August 2016; Revised: 6 October 2017; Published: 1 June 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06896950
MathSciNet: MR3807314
Digital Object Identifier: 10.1215/00127094-2017-0058

Subjects:
Primary: 53A99
Secondary: 32S45 , 34H05

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

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 8 • 1 June 2018
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