Homotopically Invisible Singular Curves

Abstract

Given a smooth manifold $M$ and a totally nonholonomic distribution $\Delta\subset TM$ of rank $d$, 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 subriemannian energy $J:\Omega(y)\to\mathbb R$, 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 $d\geq 3$ generic subriemannian structures 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 subriemannian Minimax principle and discuss some applications).