Infinitely many closed geodesic images on simply connected compact rank-one symmetric spaces

Abstract

We prove that every smooth Riemannian metric on the underlying smooth manifold of a simply connected compact rank-one symmetric space, namely $S^n$ $(n\ge2)$, $\CP^m$ $(m\ge1)$, $\HP^m$ $(m\ge1)$, or $\OP^2$ admits infinitely many geometrically distinct prime closed geodesics. Here a prime closed geodesic is a nonconstant closed geodesic which is not an iterate of a shorter one, and geometric distinctness means that the images in the manifold are pairwise different. Thus iterating a geodesic, changing its starting point, or reversing its orientation does not count as a new geodesic. No nondegeneracy assumption on the metric is imposed. These manifolds have bounded loop-space Betti numbers, so the usual homological growth criterion does not yield this conclusion. We use the length filtration together with rotation and reversal of loops. A controlled cut construction gives a uniform filling estimate in the based loop space, with a fixed increase in length independent of degree, which we transfer to free loops. We then calculate global equivariant classes associated with odd and even iterates. Assuming finitely many prime geodesic images forces incompatible requirements at a class's first critical level. The proof treats spheres and projective spaces within a common framework and does not assume a sphere infinitude theorem.