Relative Equilibria of the Curved N-Body Problem

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Exploring the shape of the universe, this monograph addresses how to measure distances in physical space, proposing that space is Euclidean for distances around 10 AU. It offers a mathematical proof confirming the flatness of space at small cosmic scales. The work extends Newtonian gravitation through the cotangent potential, originally suggested by Ernest Schering, and examines the dynamics of N point masses in non-zero constant curvature spaces. This research connects classical dynamics, non-Euclidean geometry, geometric topology, Lie groups, and polytopes.