Homotopically invisible singular curves

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).