Homotopically invisible singular curves

Abstract

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