Abstract
Given a rank-two sub-Riemannian structure (M, Ī) and a point x0āāāM, a singular curve is a critical point of the endpoint map Fā:āĪ³āā¦āĪ³ā(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 Ī³. The main goal of this paper is to show that locally around a nice singular curve Ī³, 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.
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