## Genericity results for singular curves.(English)Zbl 1102.53019

Let $$M$$ be a smooth manifold and $$\mathcal D_{m}$$, $$m \geq 2$$, be the set of rank $$m$$ distributions on $$M$$ endowed with the Whitney $$C^{\infty}$$ topology.
In the paper under review the existence of an open set $$O_{m}$$ is shown, dense in $$\mathcal D_{m}$$, so that every nontrivial singular curve of a distribution $$D$$ of $$O_{m}$$ is of minimal order and of co-rank one. In particular, this implies that for $$m \geq 3$$, every distribution of $$O_{m}$$ does not admit nontrivial rigid curves. It is also shown that for generic sub-Riemannian structures of rank $$m \geq 3$$, there do not exist nontrivial minimizing singular curves.

### MSC:

 53C17 Sub-Riemannian geometry 58A30 Vector distributions (subbundles of the tangent bundles)
Full Text: