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).
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Notes
This definition requires the choice of an inner product in each fiber (a sub-Riemannian structure on \(\Delta \)) in order to integrate the square of the norm of \(\dot{\gamma }\), but the fact of being integrable is independent of the chosen structure (we refer the reader to [2, 14, 15] for more details).
Note that, when \(\Omega (y)\) is singular it is not clear yet what a “critical point” for J should be; this will be clarified below.
We recall that \(\overrightarrow{H}\) is defined by the equation \(\iota _{\overrightarrow{H}}\omega =-dH\)
Here we chose \(\tilde{r} \) such that \(2\tilde{r}\subset W_1\) in order to guarantee that:
$$\begin{aligned} \text {clos}\left( (\phi ^u)^{-1}(U_3\times 2\tilde{r}ac/3B)\right) \subset \text {clos}\left( U_3\times \tilde{r} B\right) . \end{aligned}$$(7)We will need this property in the proof of Corollary 8.
This follows from this elementary fact. Let \(F\,{:}\,P\times K\rightarrow \mathbb {R}\) be a continuous function, where P and K are (just) topological spaces and K is compact. Define \(f(p)=\max _{k\in K}F(p,k)\). Then f is continuous (and the analogue statement with \(\max \) replaced with \(\min \) is also true). The proof is easy and left to the reader.
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Communicated by A. Malchiodi.
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Agrachev, A.A., Boarotto, F. & Lerario, A. Homotopically invisible singular curves. Calc. Var. 56, 105 (2017). https://doi.org/10.1007/s00526-017-1203-z
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DOI: https://doi.org/10.1007/s00526-017-1203-z