Starting from well-known absolute instruments that provide perfect imaging, we analyze a class of rotationally symmetric compact closed manifolds, namely, geodesic lenses. We demonstrate with a numerical method that light rays confined on geodesic lenses form closed trajectories, and that for optical waves, the spectrum of a geodesic lens is (at least approximately) degenerate and equidistant. Moreover, we fabricate two geodesic lenses in micrometer and millimeter scales and observe curved light rays along the geodesics. Our experimental setup may offer a new platform to investigate light propagation on curved surfaces.