Length product of homologically independent loops for tori

TL;DR

This paper proves that on any Riemannian torus of dimension m with unit volume, there exist m homologically independent closed geodesics whose length product is at most m^m.

Contribution

It establishes an upper bound on the length product of homologically independent geodesics in Riemannian tori, extending understanding of geometric properties of such manifolds.

Findings

Existence of m homologically independent closed geodesics

Bound on length product by m^m

Applicable to Riemannian tori of any dimension

Abstract

We prove that any Riemannian torus of dimension $m$ with unit volume admits $m$ homologically independent closed geodesics whose length product is bounded from above by $m^m$.