Counting closed geodesics on Riemannian manifolds

TL;DR

This paper constructs a locally constant function counting closed geodesics on a manifold, exploring the properties of geodesic lengths and their relation to Riemannian metrics.

Contribution

It introduces a new geodesic count function that is locally constant on the space of metrics and lengths, based on a novel weighting of geodesic sets.

Findings

Defined a weight for compact open subsets of geodesic spaces

Constructed a geodesic count function that is locally constant

Analyzed properties of geodesic length distributions

Abstract

Fix a smooth closed manifold $M$. Let $R_M$ denote the space of all pairs $(g,L)$ such that $g$ is a $C^3$ Riemannian metric on $M$ and the real number $L$ is not the length of any closed $g$-geodesics. A locally constant geodesic count function $\pi_M:R_M\rightarrow Z$ is constructed. For this purpose, the weight of compact open subsets of the space of closed $g$-geodesics is defined and investigated for an arbitrary Riemannian metric $g$.