Relative Equilibria of the Curved N-Body Problem

Exploring the shape of the universe, this monograph addresses how to measure distances in physical space, questioning whether to follow straight lines or curved paths. It presents a mathematical proof that space is Euclidean for distances around 10 AU, confirming assumptions held by physicists. The work extends classical Newtonian gravitation into non-Euclidean spaces using the cotangent potential, originally proposed by Ernest Schering. This research uncovers intriguing connections among classical dynamics, non-Euclidean geometry, geometric topology, Lie groups, and polytopes.