Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 25I (Differential Geometry)

The fundamental theorem of regular space curves states that smooth functions $\kappa (s)>0$ and $\tau (s)$ determine a unit-speed regular curve in ${\mathbb{R}}^3$ with that curvature and torsion, uniquely up to a proper Euclidean motion of Euclidean three-space. Existence follows by solving the Frenet-Serret formulas for an oriented orthonormal frame and then integrating its tangent vector.

The map $\alpha :\mathbb{R}\to \alpha (\mathbb{R})$ is a local arc-length coordinate and, when the image is closed, a covering parametrization. The restriction of $\varphi$ to the one-dimensional submanifold $\alpha (\mathbb{R})$ therefore lifts to a map $h:\mathbb{R}\to \mathbb{R}$ satisfyingBoth curves have unit speed and $\varphi$ is an isometry, so $|h^{\prime }(s)|=1$. Continuity makes the sign constant on the connected domain, hence $h^{\prime }(s)=\pm 1$ throughout andfor some constant ${\sigma }_0$.

A proper Euclidean motion of Euclidean three-space preserves both curvature and torsion. Reversal of an arc-length parameter also preserves them: in the determinant formula for torsion, both the first and third derivatives change sign. Comparing the two expressions for the transformed curve givesand, wherever $\kappa (s)>0$,

If the sign is plus and ${\sigma }_0=0$, then $\varphi$ fixes every point of the curve. A nonidentity proper Euclidean motion with any fixed points is a rotation, whose fixed set is its rotation axis. Thus $\alpha (\mathbb{R})$ lies on that axis and is a straight line.

For $R>r>0$, the closed curveis embedded on a torus and satisfiesAfter arc-length parametrization, it meets all the requirements of part (b). Its bending and twisting vary around the two waves, so neither curvature nor torsion is constant.

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.