Closed Curves with a Nondegenerate Frenet Frame

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.

Here a regular closed curve means a smooth $\gamma:S^1\to\mathbb R^n$ with $\gamma'(t)\wedge\gamma''(t)\wedge\cdots\wedge\gamma^{(n)}(t)\ne0$ everywhere. Gram--Schmidt applied to these ordered derivatives gives its Frenet frame $E(t)\in\mathrm O(n)$. The Frenet-frame length is the length of $E$ in the standard bi-invariant Riemannian metric on $\mathrm O(n)$. Embed a convex plane oval in $\mathbb R^n$ and traverse it $m$ times. Let $\mu(n)$ be the least positive such $m$ for which this multiple traversal admits arbitrarily small smooth perturbations that are regular in the above nondegeneracy sense. The source records $\mu(2)=1$ and $\mu(3)=2$. Let $\ell_2$ be the length, in the same metric, of the circle subgroup $\mathrm{SO}(2)\subset\mathrm O(n)$ rotating a fixed plane.

Statement recorded in the supplied source.

Determine $\mu(n)$ for $n>3$. Does every regular closed curve in $\mathbb R^n$ have Frenet-frame length strictly greater than $\mu(n)\ell_2$? [S2]

Short English statement

How many traversals of a plane oval permit a nearby nondegenerate curve in higher dimension, and do they determine a strict lower bound for frame length?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6500006 (AMR-064-0006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. A. Agrachev, Some open problems, arXiv:1304.2590v2 (2013), Section VI, pp. 7--8. https://arxiv.org/pdf/1304.2590.