Chitour, Y.; Jean, F.; Trélat, E. Genericity results for singular curves. (English) Zbl 1102.53019 J. Differ. Geom. 73, No. 1, 45-73 (2006). 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. Reviewer: Adrian Sandovici (Piatra Neamt) Cited in 3 ReviewsCited in 39 Documents MSC: 53C17 Sub-Riemannian geometry 58A30 Vector distributions (subbundles of the tangent bundles) Keywords:paracompact manifold; horizontal distribution; tangent bundle; singular curve; sub-Riemannian structure PDFBibTeX XMLCite \textit{Y. Chitour} et al., J. Differ. Geom. 73, No. 1, 45--73 (2006; Zbl 1102.53019) Full Text: DOI