Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 26I / c / Solution

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.