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CHORDS, LIGHT, AND ANOTHER SYNTHETIC CHARACTERIZATION OF THE ROUND SPHERE

By Benjamin Schmidt and Juan Souto

Abstract

Abstract. A chord for a closed geodesic γ in a complete Riemannian manifold M is a nontrivial geodesic segment beginning and ending on γ that is not completely contained in γ. We prove the existence of at least one geodesic chord for every closed geodesic in a closed Riemannian manifold. As an application, we give a synthetic characterization of round spheres in terms of blocking light. The study of closed geodesics in Riemannian manifolds has a long and rich history. In compact manifolds with nontrivial fundamental group, closed geodesics are at least as plentiful as free homotopy classes; namely, homotopically essential curves can be pulled tight to closed geodesics. For compact, simply connected manifolds, more sophisticated techniques are needed to prove the existence of closed geodesics. In the 1930’s Lyusternik and Shnirelman (see [Ba78]) proved that every closed simply connected manifold contains at least 3 geometricall

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