Simple closed geodesics in dimensions $\ge 3$

TL;DR

This paper proves that for generic metrics on higher-dimensional manifolds, all closed geodesics are simple and non-intersecting, leading to exponential growth in the number of such geodesics with respect to length.

Contribution

It establishes the generic simplicity and non-intersection of closed geodesics in dimensions three and higher, extending previous results and implications for geodesic counting.

Findings

All closed geodesics are simple and disjoint for generic metrics.

Number of geometrically distinct closed geodesics grows exponentially.

Results apply to both Riemannian and reversible Finsler metrics.

Abstract

We show that for a generic Riemannian or reversible Finsler metric on a compact differentiable manifold $M$ of dimension at least three all closed geodesics are simple and do not intersect each other. Using results by Contreras~\cite{C2010} \cite{C2011} this shows that for a generic Riemannian metric on a compact and simply-connected manifold all closed geodesics are simple and the number $N(t)$ of geometrically distinct closed geodesics of length $\le t$ grows exponentially.