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.

闵可夫斯基第二定理可以说成n维平面Finsler花托的不等式，它关系到封闭测地线的体积和长度的最小乘积，形成了一个同源性基础。在本文中，我们展示了如何将此基本结果提升为适用于更大类Finsler流形的原理。这包括歧管，其第一个贝蒂数和尺寸不一定要重合，主要的例子是表面。此类流形是通过杯形乘积的第一个\（\ mathbb {Z} _2 \）-同调性组上的流形基本类所诱导的多线性映射的超行列式还原模2的不消失条件来描述的。