Uniqueness of non-Euclidean Mass Center System and Generalized Pappus' Centroid Theorems in Three Geometries

TL;DR

This work addresses the non-Euclidean mass center problem by establishing a simple, axiomatic framework that yields a unique mass center system in spaces with dimension at least two and extends the concept to general manifolds. It introduces a moving-basis method to prove a highly generalized Pappus centroid theorem that simultaneously covers Euclidean, spherical, and hyperbolic geometries in arbitrary dimensions, enabling explicit volume formulas for previously unknown non-Euclidean solids such as torii and cones. The main contributions are a streamlined uniqueness proof for multi-dimensional spaces, the manifold extension with concrete centered-mass computations, and a unified, compact derivation of non-Euclidean volume formulas via Pappus’ theorem. These results provide a coherent, widely applicable framework for mass-center computation and volume calculation in non-Euclidean geometry, with potential applications in geometric analysis and mathematical physics.

Abstract

G.A. Galperin introduced the axiomatic mass center system for finite point sets in spherical and hyperbolic spaces, proving the uniqueness of the mass center system. In this paper, we revisit this system and provide a significantly simpler proof of its uniqueness. Furthermore, we extend the axiomatic mass center system to manifolds. As an application of our system, we derive a highly generalized version of Pappus' centroid theorem for volumes in three geometries - Euclidean, spherical, and hyperbolic - across all dimensions, offering unified and notably simple proofs for all three geometries.

Figures (8)

• Figure 1: Euclidean Space $\mathbb{E}^n$ lying on the ambient space $\mathbb{R}^{n+1}$, a material point $[\bm{\mathrm{a}}]$ with mass $m_{\bm{\mathrm{a}}}$, and the corresponding material vector $\bm{\mathrm{a}}$. The concepts of material point, mass, and material vector will be introduced in the following section.
• Figure 2: Spherical Space $\mathbb{S}^n$ and hyperbolic space $\mathbb{H}^n$ lying on the ambient space $\mathbb{R}^{n+1}$, material points $[\bm{\mathrm{a}}]$ with mass $m_{\bm{\mathrm{a}}}$, and the corresponding material vectors $\bm{\mathrm{a}}$
• Figure 3: concurrent segments $\overline{AF}$, $\overline{CD}$, $\overline{EG}$ and the induction hypothesis applied on the points $E$, $F$, $G$
• Figure 4: five material points on $\mathbb{S}^1$
• Figure 5: two material points and their mass center

Theorems & Definitions (43)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof