Cutting Corners in Sub-Riemannian Spaces

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. Let $\gamma_0,\gamma_1:[0,1]\to M$ be smooth horizontal paths with $\gamma_0(0)=\gamma_1(0)=q_0$ and linearly independent initial velocities, expressed as $\dot\gamma_0(0)\wedge\dot\gamma_1(0)\ne0$. The corner connection from $\gamma_0(1)$ to $\gamma_1(1)$ reverses $\gamma_0$ and then follows $\gamma_1$; its length is $L(\gamma_0)+L(\gamma_1)$.

Statement recorded in the supplied source.

Does there always exist a horizontal path from $\gamma_0(1)$ to $\gamma_1(1)$ of length strictly less than $L(\gamma_0)+L(\gamma_1)$? [S2]

Short English statement

Can every horizontal corner formed by independent tangent directions be replaced by a shorter horizontal connection?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6500002 (AMR-064-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. A. Agrachev, Some open problems, arXiv:1304.2590v2 (2013), Section II, p. 3. https://arxiv.org/pdf/1304.2590.