# Genericity results for singular curves

Abstract : Let $M$ be a smooth manifold and ${\cal D}_m$, $m\geq 2$, be the set of rank $m$ distributions on $M$ endowed with the Whitney $C^\infty$ topology. We show the existence of an open set $O_m$ dense in ${\cal D}_m$, so that, every nontrivial singular curve of a distribution $D$ of $O_m$ is of minimal order and of corank one. In particular, for $m\geq 3$, every distribution of $O_m$ does not admit nontrivial rigid curves. As a consequence, for generic sub-Riemannian structures of rank greater than or equal to three, there does not exist nontrivial minimizing singular curves.
Type de document :
Article dans une revue
Journal of Differential Geometry, 2006, 73 (1), pp.45--73

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https://hal.archives-ouvertes.fr/hal-00086357
Contributeur : Emmanuel Trélat <>
Soumis le : mardi 18 juillet 2006 - 18:58:02
Dernière modification le : jeudi 5 avril 2018 - 12:30:05
Document(s) archivé(s) le : mardi 6 avril 2010 - 00:14:58

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• HAL Id : hal-00086357, version 1

### Citation

Yacine Chitour, Frédéric Jean, Emmanuel Trélat. Genericity results for singular curves. Journal of Differential Geometry, 2006, 73 (1), pp.45--73. 〈hal-00086357〉

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