Three dimensional central configurations in H3 and S3

TL;DR

This paper investigates the properties of central configurations in three-dimensional hyperbolic and spherical spaces, revealing their equivalence to configurations in lower dimensions and identifying unique configurations in three-dimensional spheres.

Contribution

It establishes the equivalence of three-dimensional hyperbolic central configurations to two-dimensional ones and highlights the existence of unique configurations in three-dimensional spheres.

Findings

Central configurations in hyperbolic space are equivalent to those in two-dimensional hyperbolic space.

Existence of special and ordinary central configurations in three-dimensional spheres not confined to lower dimensions.

Provides a classification framework for central configurations in curved three-dimensional spaces.

Abstract

We show that each central configuration in the three-dimensional hyperbolic sphere is equivalent to one central configuration on a particular two- dimensional hyperbolic sphere. However, there exist both special and ordinary central configurations in the three-dimensional sphere that are not confined to any two-dimensional sphere.