Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 25F (Differential Geometry)

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.

For $K>0$ the number can again be infinite: remove two points away from an infinite family of great circles on the round sphere. The result is a cylinder of positive curvature containing those intersecting simple closed geodesics.

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.