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

TL;DR

This paper simplifies the proof of the uniqueness of the axiomatic mass center system in non-Euclidean geometries and extends it to manifolds, leading to a generalized Pappus' centroid theorem applicable across Euclidean, spherical, and hyperbolic spaces.

Contribution

It provides a simpler proof of the mass center system's uniqueness and extends it to manifolds, enabling a unified generalization of Pappus' centroid theorem in three geometries.

Findings

Simplified proof of mass center system uniqueness.

Extension of the system to manifolds.

Unified generalized Pappus' theorem across geometries.

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.