Homotopically invisible singular curves (Q2406069)

Let \(M\) be a smooth manifold with a totally nonholonomic distribution \(\Delta\subseteq TM\) (meaning \(\Delta\) is represented locally by vector fields whose Lie brackets generate the tangent space). The pair \((M,\Delta)\) is endowed with a sub-Riemannian structure. Given a point \(x\in M\), let \(\Omega\) denote the \textit{horizontal path space} of absolutely continuous curves \(\gamma:[0,1]\to M\) with \(\gamma(0)=x\) such that \(\dot\gamma\in \Delta\) almost everywhere and is \(L^2\) integrable. The \(W^{1,2}\) topology endows \(\Omega\) with the structure of a Hilbert manifold. Under these conditions, there is an energy functional \(J:\Omega\to \mathbb{R}\) defined by \[ J(\gamma) = \frac12 \int_{[0,1]} |\dot\gamma(t)|^2 \, dt. \] The critical points of \(J\) are called \textit{normal geodesics}, and are locally length-minimizing for the Carnot-Caratheodory distance on \(M\). The endpoint map \(F:\Omega\to M, \gamma\mapsto\gamma(1)\) is smooth, and its critical points are called \textit{singular curves}. Given \(y\in M\), define \(\Omega(y)=F^{-1}(y)\), the space of horizontal paths from \(x\) to \(y\) in \(M\); if \(y\) is a regular value of \(F\) then \(\Omega(y)\) is a smooth Hilbert manifold. The authors use a Morse-theoretic approach to study the homotopy types of the \(\Omega(y)\) in general. It was previously known that the inclusion of each \(\Omega(y)\) into the full path space is a homotopy equivalence, regardless of whether \(y\) is a regular value [the last two authors, Comm. Anal. Geom. 25 (2017), No. 2, 269--301 (2017)]; in particular each \(\Omega(y)\) is homotopy equivalent to the based loop space \(\Omega M\). Here the authors consider the (weak) homotopy type of the level sets \(\Omega(y)^E:=\{\gamma\in \Omega(y) \mid J(\gamma)\leq E\}\) of the energy functional, and prove the following ``sub-Riemannian deformation Lemma''. Theorem 1. Assume that all singular curves with energy \(J\leq E_2\) are soft and that there are no normal geodesics in \(\Omega(y)\) with energy in \([E_1,E_2]\). Then for every compact manifold \(X\) and \(\varepsilon>0\), any continuous map \(h:X\to \Omega(y)^{E_2}\) can be deformed into \(\Omega(y)^{E_1 +\varepsilon}\). In particular, \(\Omega(y)^{E_2}\) and \(\Omega(y)^{E_1+\varepsilon}\) are weakly homotopy equivalent. A singular curve is \textit{soft} if it is (a) of corank one, (b) not Goh, and (c) strictly abnormal. When the rank of the distribution is at least 3, the sub-Riemannian structures whose singular curves are all soft are generic, by a result of [\textit{Y. Chitour} et al., J. Differ. Geom. 73, No. 1, 45--73 (2006; Zbl 1102.53019)]. The proof would be difficult to summarize here, but involves passing from the space of curves to the space of controls, and showing that a corresponding endpoint map has the Serre fibration property. Applications are given to the calculus of variations on the singular space of horizontal curves (a ``sub-Riemannian minimax principle'' is proved) and the existence of infinitely many normal geodesics.