Abstract
Given a rank-two sub-Riemannian structure (M, Î) and a point x0 â M, a singular curve is a critical point of the endpoint map F : Îł ⊠γ (1) defined on the space of horizontal curves starting at x0. The typical least degenerate singular curves of these structures are called regular singular curves; they are nice if their endpoint is not conjugate along Îł. The main goal of this paper is to show that locally around a nice singular curve Îł, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which F writes essentially as a sum of a linear map and a quadratic form. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have