Given a smooth manifold M and a totally nonholonomic distribution Δ⊂TMΔ⊂TM of rank d≥3d≥3 , we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M. Singular curves are critical points of the endpoint map F:γ↦γ(1)F:γ↦γ(1) defined on the space ΩΩ of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy J:Ω(y)→RJ:Ω(y)→R , where Ω(y)=F−1(y)Ω(y)=F−1(y) is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets {γ∈Ω(y)|J(γ)≤E}{γ∈Ω(y)|J(γ)≤E} . We call them homotopically invisible. It turns out that for d≥3d≥3 generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73, 2006) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications).

Homotopically invisible singular curves / Agrachev, Andrey; Boarotto, Francesco; Lerario, Antonio. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 56:4(2017), pp. 1-34. [10.1007/s00526-017-1203-z]

Homotopically invisible singular curves

Agrachev, Andrey;Boarotto, Francesco;Lerario, Antonio
2017-01-01

Abstract

Given a smooth manifold M and a totally nonholonomic distribution Δ⊂TMΔ⊂TM of rank d≥3d≥3 , we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M. Singular curves are critical points of the endpoint map F:γ↦γ(1)F:γ↦γ(1) defined on the space ΩΩ of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy J:Ω(y)→RJ:Ω(y)→R , where Ω(y)=F−1(y)Ω(y)=F−1(y) is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets {γ∈Ω(y)|J(γ)≤E}{γ∈Ω(y)|J(γ)≤E} . We call them homotopically invisible. It turns out that for d≥3d≥3 generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73, 2006) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications).
2017
56
4
1
34
105
https://arxiv.org/abs/1603.08937
Agrachev, Andrey; Boarotto, Francesco; Lerario, Antonio
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/59084
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