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.
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Submitted on : Tuesday, July 18, 2006 - 6:58:02 PM
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Yacine Chitour, Frédéric Jean, Emmanuel Trélat. Genericity results for singular curves. Journal of Differential Geometry, 2006, 73 (1), pp.45-73. ⟨10.4310/jdg/1146680512⟩. ⟨hal-00086357⟩

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