Unfolding the Sub-Riemannian Distance

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. A distance sphere centered at $q_0$ is $\{q:d_D(q_0,q)=r\}$. A germ records its shape in arbitrarily small neighborhoods of a chosen point; $C^1$ equivalence is equivalence by a local ambient $C^1$ diffeomorphism. The points of interest lie on length-minimizing singular curves. The Martinet case is a rank-two distribution in $\mathbb R^3$ whose bracket growth drops to $(2,2,3)$ on its Martinet surface; the Engel case is the rank-two Engel distribution in $\mathbb R^4$ (growth $(2,3,4)$). The metric on the given distribution varies generically. The same section also asks about a two-dimensional almost-Riemannian model. Let smooth fields $f_0,f_1$ satisfy $f_0(q_0)\wedge f_1(q_0)=0$, $n_2(q_0)=1$ and $n_3(q_0)=2$, where $n_j$ is the dimension of the span of brackets of length at most $j$. Admissible velocities are $u_0f_0+u_1f_1$ with $u_0^2+u_1^2=1$, and the induced distance is minimum travel time.

Statement recorded in the supplied source.

Find a $C^1$ classification of sphere germs at points of optimal singular curves for generic sub-Riemannian metrics, particularly generic Martinet metrics in $\mathbb R^3$ and the Engel distribution in $\mathbb R^4$. In the two-dimensional model just specified, classify the germs at $q_0$ of the associated distance function for generic pairs $(f_0,f_1)$. [S2]

Short English statement

Classify the local shapes of distance spheres along minimizing abnormal curves, including the Martinet and Engel models and the related two-dimensional degenerate model.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6500004 (AMR-064-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. A. Agrachev, Some open problems, arXiv:1304.2590v2 (2013), Section IV, pp. 4--6. https://arxiv.org/pdf/1304.2590.