Normal forms for the endpoint map near nice singular curves for rank-two distributions

Agrachev, Andrei A.; Boarotto, Francesco

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. S30

Given a rank-two sub-Riemannian structure (M, Δ) and a point x₀ ∈ M, a singular curve is a critical point of the endpoint map F : γ ↦ 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 γ. The main goal of this paper is to show that locally around a nice singular curve γ, 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.

Classification : 53C17, 58K50, 58K05
Keywords: Sub-Riemannian geometry, abnormals, endpoint mapping, normal forms

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