Length spectrum of geodesic loops in manifolds of nonpositive curvature

TL;DR

This paper establishes a lower bound on the number of distinct geodesic loops of bounded length in nonpositively curved manifolds, linking it to volume growth and extending classical results on geodesic loops.

Contribution

It reformulates Blichfeldt's theorem within nonpositive curvature manifolds to derive new bounds on geodesic loops based on volume growth.

Findings

Lower bound depends on volume growth and manifold volume

Comparison with known asymptotic geodesic growth results

Extension of classical theorems to nonpositive curvature settings

Abstract

In section 1 we reformulate a theorem of Blichfeldt in the framework of manifolds of nonpositive curvature. As a result we obtain a lower bound on the number of homotopically distinct geodesic loops emanating from a common point q whose length is smaller than a fixed constant. This bound depends only on the volume growth of balls in the universal covering and the volume of the manifold itself. We compare the result with known results about the asymptotic growth rate of closed geodesics and loops in section 2.