  The question of whether a closed Riemannian manifold has infinitely many
geometrically distinct closed geodesics has a long history. Though unsolved in
general, it is well understood in the case of surfaces. For surfaces of
revolution diffeomorphic to the sphere, a refinement of this problem was
introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this
article, we quantify their result by counting distinct geodesics of bounded
length. In addition, we reframe these results to obtain a couple of
characterizations of the round two-sphere.
