Homotopically Invisible Singular Curves

TL;DR

This paper investigates the impact of singular curves on the topology of horizontal path spaces in subriemannian manifolds, showing that for dimensions three and higher, most singular curves are homotopically invisible, which simplifies the calculus of variations in this context.

Contribution

It introduces the concept of homotopically invisible singular curves and proves that generic subriemannian structures in dimensions three and above have only such curves, advancing the calculus of variations on singular spaces.

Findings

Most singular curves are homotopically invisible in generic structures for d≥3.

Homotopically invisible singular curves do not affect the topology of energy sublevel sets.

A Minimax principle for subriemannian energy on singular spaces is established.

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).