Morse-Sard Questions for Endpoint Maps

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A sub-Riemannian structure consists of a smooth bracket-generating distribution $D\subset TM$ and a smooth inner product on $D$. A horizontal (admissible) curve has $\dot\gamma(t)\in D_{\gamma(t)}$ almost everywhere, and length $L(\gamma)=\int\|\dot\gamma(t)\|_D\,dt$. The distance is the infimum of lengths of horizontal connections. Fix $q_0\in M$ and let $\mathcal H_{q_0}$ be the Hilbert manifold of $H^1$ horizontal paths $\gamma:[0,1]\to M$ starting at $q_0$. Its endpoint map is $\mathcal E(\gamma)=\gamma(1)$. A curve is singular (abnormal) if the differential $D_\gamma\mathcal E$ is not surjective onto $T_{\gamma(1)}M$. It is optimal singular if it is also length minimizing between its endpoints. Positive measure refers to the smooth measure class on $M$, equivalently any smooth positive volume density.

Statement recorded in the supplied source.

Can the endpoints of all singular curves starting at $q_0$ fill all of $M$? Can the endpoints of optimal singular curves starting at $q_0$ fill a subset of positive measure in $M$? [S2]

Short English statement

How large can the endpoint sets of abnormal curves and of minimizing abnormal curves be?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6500003 (AMR-064-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. A. Agrachev, Some open problems, arXiv:1304.2590v2 (2013), Section III, pp. 3--4. https://arxiv.org/pdf/1304.2590.