Infinitely many closed geodesics on every Riemannian sphere and every closed three-manifold

Theorem 1.1: every smooth Riemannian metric on $S^n$, $n\ge2$, has infinitely many prime closed geodesics with pairwise distinct images. Corollary 1.2: the same for every closed manifold finitely covered by a sphere (real projective spaces, spherical space forms). Corollary 1.3: the same for every closed three-manifold, orientable or not, using Rademacher-Taimanov and Perelman. It does NOT settle the problem for general closed manifolds in dimension four or more, and it gives no growth rate or length count for the geodesics.

Theorem 1.1: every smooth Riemannian metric on $S^n$, $n\ge2$, has infinitely many prime closed geodesics with pairwise distinct images. Corollary 1.2: the same for every closed manifold finitely covered by a sphere (real projective spaces, spherical space forms). Corollary 1.3: the same for every closed three-manifold, orientable or not, using Rademacher-Taimanov and Perelman. It does NOT settle the problem for general closed manifolds in dimension four or more, and it gives no growth rate or length count for the geodesics.