Closed geodesics on connected sums and 3-manifolds

TL;DR

This paper investigates the growth of the number of distinct closed geodesics on connected sums of manifolds, showing that generic metrics on 3-manifolds have infinitely many such geodesics, with growth comparable to prime numbers.

Contribution

It establishes new asymptotic growth results for closed geodesics on connected sums of manifolds, linking geometric properties to prime number growth patterns.

Findings

Number of closed geodesics grows at least like prime numbers on certain 3-manifolds.

Generic Riemannian metrics on compact 3-manifolds have infinitely many closed geodesics.

Results apply to prime decomposition of 3-manifolds with non-trivial fundamental groups.

Abstract

We study the asymptotics of the number N(t) of geometrically distinct closed geodesics 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 this result 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 simply-connected manifold which is not homeomorphic to a sphere.