A Closing Lemma for Flat Strips in Compact Surfaces of Non-positive Curvature

Let M be a compact oriented Riemannian surface with non-positive curvature. Suppose that F : R× [0, ε] → M is an isometric immersion. We show that either M is isometric to flat torus or the image of F is periodic, i.e., it is a periodic flat strip in M. Introduction The results of this paper are related to a well-known problem to the effect that the geodesic flow on the unit tangent bundle of a surface of non-positive curvature and genus ≥ 2 must be ergodic. This conjecture remains widely open. If a compact, non-positively curved Riemannian surface (M, g) has a flat strip, its geodesic flow is not necessarily hyperbolic in the sense of Anosov. Therefore, the ergodicity problem is related to the study the distribution of zero curvatures on the surface. We shall show the following result Main Theorem. Let (M, g) be a compact surface without boundary. Suppose that the curvature is non-positive and F : [0,∞)× [0, ε] → M is an isometric immersion. Then either (M, g) is isometric to flat torus or the image of F is periodic, i.e., it is a periodic flat strip in M. We should point out that for any given flat torus (T , g), there exist examples of total geodesic isometric immersions F of flat strips such that F is not periodic. For instance, let (T , g0) be a flat square torus which is isometric to S1×S1. If we choose a flat strip F with irrational slope, then the image of F is dense in T . However, such a strip is not periodic. 1The work of both authors was partially supported by NSF grants.