# Homotopically invisible singular curves

Homotopically invisible singular curves Given a smooth manifold M and a totally nonholonomic distribution $$\Delta \subset TM$$ Δ ⊂ T M of rank $$d\ge 3$$ 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 $$F\,{:}\,\gamma \mapsto \gamma (1)$$ F : γ ↦ γ ( 1 ) defined on the space $$\Omega$$ Ω of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy $$J\,{:}\,\Omega (y)\rightarrow \mathbb R$$ J : Ω ( y ) → R , where $$\Omega (y)=F^{-1}(y)$$ 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 $$\{\gamma \in \Omega (y)\,|\,J(\gamma )\le E\}$$ { γ ∈ Ω ( y ) | J ( γ ) ≤ E } . We call them homotopically invisible. It turns out that for $$d\ge 3$$ 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# Homotopically invisible singular curves

, Volume 56 (4) – Jul 10, 2017
34 pages

/lp/springer_journal/homotopically-invisible-singular-curves-gMGrzNu9nn
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1203-z
Publisher site
See Article on Publisher Site

### Abstract

Given a smooth manifold M and a totally nonholonomic distribution $$\Delta \subset TM$$ Δ ⊂ T M of rank $$d\ge 3$$ 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 $$F\,{:}\,\gamma \mapsto \gamma (1)$$ F : γ ↦ γ ( 1 ) defined on the space $$\Omega$$ Ω of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy $$J\,{:}\,\Omega (y)\rightarrow \mathbb R$$ J : Ω ( y ) → R , where $$\Omega (y)=F^{-1}(y)$$ 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 $$\{\gamma \in \Omega (y)\,|\,J(\gamma )\le E\}$$ { γ ∈ Ω ( y ) | J ( γ ) ≤ E } . We call them homotopically invisible. It turns out that for $$d\ge 3$$ 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).

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jul 10, 2017

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