| Issue |
ESAIM: COCV
Volume 27, 2021
Regular 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
|
|
|---|---|---|
| Article Number | S30 | |
| Number of page(s) | 32 | |
| DOI | https://doi.org/10.1051/cocv/2020089 | |
| Published online | 01 March 2021 | |
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: This email address is being protected from spambots. You need JavaScript enabled to view it.
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.
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”.
F.B. and A.A.A. warmly thank the anonymous reviewer for the care paid in the revision process. Many parts of this text have been significantly polished and simplified thanks to his/her suggestions.
© EDP Sciences, SMAI 2021
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