Duke Mathematical Journal

The Sard conjecture on Martinet surfaces

André Belotto da Silva and Ludovic Rifford

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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

Export citation


  • [1] A. Agrachev, “Some open problems” in Geometric Control Theory and Sub-Riemannian Geometry (Cortona, 2012), Springer INdAM Ser. 5, Springer, Cham, 2014, 1–13.
  • [2] A. Agrachev, A. Barilari, and U. Boscain, Introduction to Riemannian and sub-Riemannian Geometry, in preparation.
  • [3] S. M. Bates and C. G. Moreira, De nouvelles perspectives sur le théorème de Morse-Sard, C. R. Math. Acad. Sci. Paris 332 (2001), 13–17.
  • [4] P. Baum and R. Bott, Singularities of holomorphic foliations, J. Differential Geom. 7 (1972), 279–342.
  • [5] A. Bellaïche, “The tangent space in sub-Riemannian geometry” in Sub-Riemannian Geometry, Progr. Math. 144, Birkhäuser, Basel, 1996, 1–78.
  • [6] E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 5–42.
  • [7] E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207–302.
  • [8] E. Bierstone and P. D. Milman, Functoriality in resolution of singularities, Publ. Res. Inst. Math. Sci. 44 (2008), 609–639.
  • [9] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. of Math. (2) 79 (1964), 109–203 and 205–326.
  • [10] H. Hironaka, Introduction to Real-analytic Sets and Real-analytic Maps, Istituto Matematico “L. Tonelli” dell’Università di Pisa, Pisa, 1973.
  • [11] J. Kollár, Lectures on Resolution of Singularities, Ann. of Math. Stud. 166, Princeton Univ. Press, Princeton, 2007.
  • [12] C. Kottke and R. Melrose, Generalized blow-up of corners and fiber products, Trans. Amer. Math. Soc. 367 (2015), 651–705.
  • [13] W. Liu and H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564.
  • [14] S. Lojasiewicz, “Sur les ensembles semi-analytiques” in Actes du congrès international des mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, 237–241.
  • [15] J. W. Milnor, Topology From the Differentiable Viewpoint, Princeton Landmarks in Math., Princeton Univ. Press, Princeton, 1997.
  • [16] R. Montgomery, A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr. 91, Amer. Math. Soc. Providence, 2002.
  • [17] R. Narasimhan, Introduction to the Theory of Analytic Spaces, Lecture Notes in Math. 25, Springer, Berlin, 1966.
  • [18] D. Panazzolo, Resolution of singularities of real-analytic vector fields in dimension three, Acta Math. 197 (2006), 167–289.
  • [19] L. Rifford, Sub-Riemannian Geometry and Optimal Transport, Springer, Cham, 2014.
  • [20] L. Rifford, Singulières minimisantes en géométrie sous-Riemannienne, Astérisque 390 (2017), 277–301.
  • [21] L. Rifford and E. Trélat, Morse-Sard type results in sub-Riemannian geometry, Math. Ann. 332 (2005), 145–159.
  • [22] H. J. Sussmann, “A cornucopia of four-dimensional abnormal sub-Riemannian minimizers” in Sub-Riemannian Geometry, Progr. Math. 144, Birkhäuser, Basel, 1996, 341–364.
  • [23] A. Tognoli, “Some results in the theory of real analytic spaces” in Espaces analytiques (Bucharest, 1969), Editura Acad. R.S.R., Bucharest, 1971, 149–157.
  • [24] J. Tougeron, Idéaux de fonctions différentiables, Ergeb. Math. Grenzgeb. (3) 71, Springer, Berlin, 1972.
  • [25] R. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Pure Appl. Math. 43, Marcel Dekker, New York, 1977.
  • [26] J. Włodarczyk, “Resolution of singularities of analytic spaces” in Geometry-Topology Conference (Gökova, 2008), International Press, Somerville, Mass., 2009, 31–63.
  • [27] I. Zelenko and M. Zhitomirskii, Rigid paths of generic 2-distributions on 3-manifolds, Duke Math. J. 79 (1995), 281–307.