Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the kth moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.

透镜空间是一类流形，它们是拓扑和微分几何中许多有趣现象的来源。它们的具体构造作为有限循环群的自由线性作用的奇数维球商，可以对其结构进行更深入的分析。在本文中，我们考虑了齐次三维透镜空间上随机选择的点对之间的距离函数的矩问题。我们给出了矩的递归关系，第k矩的公式和矩产生函数的公式，以及这些透镜空间中所有半径的球的体积的显式公式。