Closed geodesics on connected sums and $3$-manifolds

Abstract

We study the asymptotics of the number $N(t)$ of geometrically distinct closed geodesics of length $\leq t$ of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups, and apply the results to the prime decomposition of a three-manifold. In particular we show that the function $N(t)$ grows at least like the prime numbers on a compact $3$-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact $3$-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simplyconnected manifold which is not homeomorphic to a sphere.