Infinitely many closed geodesic images on every Riemannian sphere

We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere S${}^n$, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.