Genericity results for singular curves

Abstract

Let M be a smooth manifold and D_m, m ≥ 2, be the set of rank m distributions on M endowed with the Whitney C^∞ topology. We show the existence of an open set O_m dense in 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 > 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 do not exist nontrivial minimizing singular curves.