My problem:
Suppose $\sigma : I \to \mathbb{R}^3$ is a biregular curve parametrized by arc length, having constant curvature $k_0>0$.
I would like to prove that the support of $\sigma$ is contained in a sphere with radius $R>0$ if and only if $k_0 > \frac 1 R$ and torsion $\tau$ is zero.
My attempt:
I observed that if $\|\sigma(s)\|=R$ then $\langle \sigma,\sigma' \rangle=0$, $\langle \sigma'',\sigma' \rangle=0$ and $\langle \sigma''',\sigma'' \rangle=0$ but I do not know how to continue.