Minimal length product over homology bases of manifolds

TL;DR

This paper extends Minkowski's second theorem, relating volume and minimal length products of homology basis geodesics, to a broader class of Finsler manifolds, including surfaces with differing Betti number and dimension.

Contribution

It generalizes a classical inequality from flat Finsler tori to a wider class of Finsler manifolds using a hyperdeterminant condition.

Findings

The inequality holds for manifolds with non-vanishing hyperdeterminant mod 2.

Includes manifolds where Betti number and dimension differ.

Applicable to surfaces with complex topological properties.

Abstract

Minkowski's second theorem can be stated as an inequality for $n$-dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology basis. In this paper we show how this fundamental result can be promoted to a principle holding for a larger class of Finsler manifolds. This includes manifolds for which first Betti number and dimension do no necessarily coincide, a prime example being the case of surfaces. This class of manifolds is described by a non-vanishing condition for the hyperdeterminant reduced modulo $2$ of the multilinear map induced by the fundamental class of the manifold on its first ${\mathbb Z}_2$-cohomology group using the cup product.