On the Existence of Closed Geodesics on Two-Spheres

In [7] J. Franks proves the existence of infinitely many closed geodesics for every Riemannian metric on S 2 which satisfies the following condition: there exists a simple closed geodesic for which Birkhoff's annulus map is defined. In particular, all metrics with positive Gaussian curvature have this property. Here we prove the existence of infinitely many closed geodesics for every Riemannian metric on S 2 which has a simple closed geodesic for which Birkhoff's annulus map is not defined. Combining this with J. Franks's result and with the fact that every Riemannian metric on S 2 has a simple closed geodesic one obtains the existence of infinitely many closed geodesics for every Riemannian metric on S 2 .