Preface

The guiding light of this monograph is a question easy to understand but difficult to answer: What is the shape of the universe? In other words, how do we measure the shortest distance between two points of the physical space? Should we follow a straight line, as on a flat table, fly along a great circle of a sphere, as between Paris and New York, or take some other course, and if so, what would that path look like? If we accept the model proposed here, which assumes that a Newtonian gravitational law extended to a universe of constant curvature is a good approximation of the physical reality (and we will later outline a few arguments in favor of this approach), then we can hint at a potential proof to the above question for distances comparable to those of our solar system. More precisely, this monograph provides a first step towards showing that, for distances of the order of 10AU, space is Euclidean. Even if rigorously proved, this conclusion won’t surprise astronomers, who accept the small-scale flatness of the universe due to the many observational confirmations they have. But the analysis of some recent spaceship orbits raises questions either about the geometry of space or our understanding of gravitation, [26]. However, we cannot emphasize enough that the main goal of this monograph is mathematical. We aim to shed some light on the dynamics of N point masses that move in spaces of nonzero constant curvature according to an attraction law which extends classical Newtonian gravitation beyond R3. This natural generalization employs the cotangent potential, first introduced in 1870 by Ernst Schering, who obtained its analytic expression following the geometric approach of János Bolyai and Nikolai Lobachevsky for a 2-body problem in hyperbolic space, [87], [8], [70]. As Newton’s idea of gravitation was to use a force inversely proportional to the area of a sphere of radius equal in length to the Euclidean distance between the bodies, Bolyai and Lobachevsky thought of a similar definition in terms of the hyperbolic distance in hyperbolic space. Our generalization of the cotangent potential to any number N of bodies led us to the recent discovery of some interesting properties, [35], [36]. These new results reveal certain connections among at