Appears in collection : Journée sous-riemannienne 2018

Let (M, ∆) be a rank-two sub-Riemannian structure on a smooth manifold M, and let x, y be any two points on M. In this talk I will present some recent results concerning the description of the set Ω(y), of all the horizontal curves joining x and y, in the vicinity of a rank-two-nice singular curve γ. This is made possible by the existence of a normal form for the endpoint map F locally around γ, and in turn this result permits to discuss some rather surprising isolation properties of γ among extremal curves. If time permits, we will try to discuss some topological properties of rank-two-nice singular curves, establishing in particular their homotopical visibility. This is a joint work with A. Agrachev and A. Lerario.