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Homotopically invisible singular curves

Given a smooth manifold M and a totally nonholonomic distribution Δ ⊂ T M of rank d ≥ 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 ) defined on the space Ω o...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2017-08, Vol.56 (4), p.1-34, Article 105
Main Authors: Agrachev, Andrei A., Boarotto, Francesco, Lerario, Antonio
Format: Article
Language:English
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Summary:Given a smooth manifold M and a totally nonholonomic distribution Δ ⊂ T M of rank d ≥ 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 ) defined on the space Ω of horizontal paths starting at a fixed point x . We consider a sub-Riemannian energy J : Ω ( y ) → R , where 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 } . We call them homotopically invisible . It turns out that for d ≥ 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).
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-017-1203-z