Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||32|
|Published online||01 March 2021|
SISSA (Trieste) and Steklov Institute,
2 Dipartimento di Matematica Tullio Levi-Civita, Università degli studi di Padova, Padova, Italy.
*** Corresponding author: firstname.lastname@example.org
Accepted: 30 November 2020
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.
Mathematics Subject Classification: 53C17 / 58K50 / 58K05
Key words: Sub-Riemannian geometry / abnormals / endpoint mapping / normal forms
F.B. has been supported by the ANR SRGI (reference ANR-15-CE40-0018), and by the University of Padova STARS Project “Sub-Riemannian Geometry and Geometric Measure Theory Issues: Old and New”.
© EDP Sciences, SMAI 2021
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