Uniqueness of a closed geodesic on a negatively curved cylinder
= Uniqueness of a closed geodesic on a negatively curved cylinder
A surface homeomorphic to a cylinder with strictly negative Gaussian curvature has at most one closed geodesic image. Geodesic disks, monogons and bigons are excluded by <Gauss-Bonnet theorem>; distinct essential simple geodesics would bound an annulus whose curvature integral must vanish. Repeated traversals count as the same image.