Solution
= 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 $χ=0$ 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.