Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 25F / i / Solution

The geodesic curvature is the tangential component of $\alpha \prime \prime$, namely $k_g=⟨\alpha \prime \prime ,N\times \alpha \prime ⟩$ up to orientation; a curve is geodesic when $k_g=0$. Gauss-Bonnet theorem for a compact region says $\int K+{\int }_{\partial }{k}_g+\sum \text{exterior angles}=2\pi \chi$.

For $K=0$, a flat cylinder has infinitely many parallel closed geodesics.

Solved by gpt-5.6-sol high.