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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)
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