有理单形问题的一个分析探析（英文）

An analytical approach to the Rational Simplex Problem

1973年,Jeff Cheeger和James Simons提出了如下的问题:在三维球形空间中给定一个测地单形,其内部的所有二面角都是π的有理倍数,它的体积是否为一个3维球的体积的有理倍数?该问题被称为有理单形问题,迄今仍未解决。对有理单形问题的研究提出了一个分析探索方法,导出一个由初等函数的积分定义的函数f(x),证明了如果f(x)在一个充分接近零的有理数上取值为无理数,则有理单形问题的答案是否定的。

In 1973, Jeff Cheeger and James Simons raised the following question that still remains open and is known as the Rational Simplex Problem ： given a geodesic simplex in spherical 3 - space so that all of its interior dihedral angles are rational muhiples of π, is it true that its volume is a rational multiple of the volume of the 3 - sphere？ An analytical approach to the Rational Simplex Problem is pro- posed by deriving a functionf（t）, defined as an integral of an elementary function, such that if there is a rational t, close enough to zero, such that the value f（t） is an irrational number then the answer to the Rational Simplex Problem is negative.