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

For $K<0$ there is at most one. Two essential simple closed geodesics cannot form a geodesic bigon by Gauss-Bonnet theorem, so they are disjoint; the annulus between them would have $\chi =0$ and geodesic boundary, forcing $\int K=0$, a contradiction. Thus multiple geodesics in the flat case are disjoint, while in the positive example they need not be.

Solved by gpt-5.6-sol high.