Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 26I (Differential Geometry)

Curvature requires a $C^2$ regular curve; torsion requires $C^3$ and nonzero curvature. The parameter-independent formulas are

For unit speed on an oriented surface,These coefficients are normal and geodesic curvature; the curve is geodesic iff ${\kappa }_g=0$.

A line segment has zero normal curvature in its tangent direction, impossible when Gaussian curvature is positive because both principal curvatures have the same nonzero sign. Two disjoint closed geodesics would bound an annulus. Applying Gauss-Bonnet theorem, its geodesic boundary contributes zero and $\chi =0$, but $\int KdA>0$, a contradiction.

If a regular $C^3$ extension existed at either finite endpoint, reparametrization by arc length would extend the Frenet data. Because $\kappa \ge \epsilon$, the denominator in the torsion formula stays away from zero; compactness of a short closed parameter interval would then bound both $\kappa$ and $|\tau |$. This contradicts the stated limsup at that endpoint. Applying the argument at both ends proves nonextendibility.