The minimal length product over homology bases of manifolds

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