Geodesic families characterizing flat metrics on a cylinder and a plane

Abstract

We prove that a complete non-compact surface contains a domain which is isometric to a pipe cylinder if all prime closed geodesics in it have the same length. As an application, we show that a flat cylinder is conjugacy rigid in the class of surfaces whose universal covering planes satisfy the divergence property. We study the divergence property from the view point of geodesic conjugacy for the Euclidean plane.