Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 25I / d / Solution

LetThis is another unit-speed embedded curve. The assumed identities say that $\alpha$ and $\beta$ have the same positive curvature and the same torsion. By the uniqueness part of the fundamental theorem of regular space curves, there is a proper Euclidean motion $\varphi$ such thatSince $s\mapsto \pm s+{\sigma }_0$ is a bijection of $\mathbb{R}$, this motion preserves $\alpha (\mathbb{R})$.

It is nontrivial. For the plus sign, ${\sigma }_0\ne 0$ and $\varphi =\mathrm{id}$ would imply $\alpha (s)=\alpha (s+{\sigma }_0)$, contradicting embeddedness. For the minus sign, the identity would imply $\alpha (s)=\alpha (-s+{\sigma }_0)$ for every $s$, and injectivity would force $s=-s+{\sigma }_0$ for every $s$, which is impossible. Thus the required nontrivial proper Euclidean motion exists.