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

¹ SISSA (Trieste) and Steklov Institute, Moscow, Russia.
² Dipartimento di Matematica Tullio Levi-Civita, Università degli studi di Padova, Padova, Italy.

^*** Corresponding author: francesco.boarotto@gmail.com

Received: 23 July 2019
Accepted: 30 November 2020

Abstract

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.