Frenet turns

Abstract

We discuss a problem posed by A.~Agrachev asking how many times a usual circle in $\mathbb R^n$ should be traversed to admit a deformation by curves with nowhere degenerating Frenet frame. It turns out that the answer depends on a specific topology which we consider. For the literal $C^n$ curve topology, the least number of turns of a plane circle admitting arbitrarily small nondegenerate perturbations is \[ k(2)=1,\qquad k(3)=2,\qquad k(n)=1\quad(n\ge4). \] This jet-level problem is different from the original Frenet-control problem by Agrachev. We show that in the literal interpretation of Agrachev's problem one has a simple spherical Fenchel obstruction in all dimensions $n\ge4$. To retain a nontrivial turn-counting problem, we introduce decorated turn data. In $\R^4$ the datum is a pair $(p,q)$ recording tangent-plane and normal-plane turns; we prove that every nonresonant pair $(p,q)$ with $p,q>0$, $p\ne q$, is accessible by small positive constant Frenet controls. In even dimension $2r$ the analogous datum is a vector $(p_1,\ldots,p_r)$, and every vector with pairwise distinct positive entries is accessible by constant controls. Odd dimensions require genuinely time-dependent openings since constant controls cannot close the base curve.