Sard conjecture for smooth bracket-generating distributions
Let $M$ be a smooth $n$-manifold and let smooth vector fields $X_1, \dots, X_r$ span a constant-rank distribution $\Delta \subset TM$ whose iterated Lie brackets span $T_xM$ at every $x$. For $x_0 \in M$ the endpoint map is $$E_{x_0} : u \longmapsto x_u(1), \qquad \dot x_u(t) = \sum_{j=1}^r u_j(t) X_j(x_u(t)), \quad x_u(0) = x_0,$$ on the open set of $u \in L^2([0,1]; \mathbb{R}^r)$ for which the trajectory exists on $[0,1]$. A control $u$ is singular when $DE_{x_0}(u)$ is not onto $T_{E_{x_0}(u)}M$. Zelenko and Zhitomirskii proved a strong form of the statement below for generic rank-two distributions on three-manifolds, and Belotto da Silva, Figalli, Parusiński and Rifford for real-analytic ones; Belotto da Silva, Parusiński and Rifford proved it for rank-three distributions on four-manifolds and for generic distributions of corank one.
Problem. Is the set of endpoints of all singular controls null for every smooth positive volume density on $M$, for every such distribution and every $x_0$?
Source: Zhitomirskii and Montgomery; see Montgomery (2002), §10.2, Rifford and Trélat (2005), §4.3, and Belotto da Silva, Parusiński and Rifford (2025), §1.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- OC
- Keywords
- abnormal curves, AIM Open Applied Problems, endpoint maps, sub-Riemannian geometry
References
@book{montgomery2002,
author = {Montgomery, Richard},
title = {A Tour of Subriemannian Geometries, Their Geodesics and Applications},
series = {Mathematical Surveys and Monographs},
volume = {91},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2002},
doi = {10.1090/surv/091},
}
@article{rifford2005,
author = {Rifford, L. and Tr{\'e}lat, E.},
title = {Morse--{S}ard type results in sub-{R}iemannian geometry},
journal = {Math. Ann.},
volume = {332},
number = {1},
year = {2005},
pages = {145--159},
doi = {10.1007/s00208-004-0622-2},
}
@article{belotto2025,
author = {Belotto da Silva, Andr{\'e} and Parusi{\'n}ski, Adam and Rifford, Ludovic},
title = {Abnormal singular foliations and the {S}ard conjecture for generic co-rank one distributions},
journal = {Rev. Mat. Iberoam.},
volume = {41},
number = {5},
year = {2025},
pages = {1599--1628},
doi = {10.4171/RMI/1567},
eprint = {2310.20284},
archivePrefix = {arXiv},
}
@article{zelenko1995,
author = {Zelenko, I. and Zhitomirskii, M.},
title = {Rigid paths of generic 2-distributions on 3-manifolds},
journal = {Duke Math. J.},
volume = {79},
number = {2},
year = {1995},
pages = {281--307},
doi = {10.1215/S0012-7094-95-07907-1},
}
@article{belotto2022,
author = {Belotto da Silva, A. and Figalli, A. and Parusi{\'n}ski, A. and Rifford, L.},
title = {Strong {S}ard conjecture and regularity of singular minimizing geodesics for analytic sub-{R}iemannian structures in dimension 3},
journal = {Invent. Math.},
volume = {229},
number = {1},
year = {2022},
pages = {395--448},
doi = {10.1007/s00222-022-01111-2},
eprint = {1810.03347},
archivePrefix = {arXiv},
}
@misc{aim-sard-conjecture-endpoint-map,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sard conjecture for smooth bracket-generating distributions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/123-sard-conjecture-endpoint-map.md},
note = {Entry 123 at commit a153f46, 2026-10-13},
}
Resolution claims
We construct an endpoint map with no regular values, disproving the sub-Riemannian Sard conjecture. Remarkably, in our construction the minimizing Sard property remains valid.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: arXiv:2609.05991v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00497, "Sard conjecture for smooth bracket-generating distributions", clopen.solutions.