Memoirs of the American Mathematical Society 2014; 80 pp; softcover Volume: 228 ISBN10: 0821891367 ISBN13: 9780821891360 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/228/1071
 The author considers the \(3\)dimensional gravitational \(n\)body problem, \(n\ge 2\), in spaces of constant Gaussian curvature \(\kappa\ne 0\), i.e. on spheres \({\mathbb S}_\kappa^3\), for \(\kappa>0\), and on hyperbolic manifolds \({\mathbb H}_\kappa^3\), for \(\kappa<0\). His goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. He also briefly discusses the issue of singularities in order to avoid impossible configurations. He derives the equations of motion and defines six classes of relative equilibria, which follow naturally from the geometric properties of \({\mathbb S}_\kappa^3\) and \({\mathbb H}_\kappa^3\). Then he proves several criteria, each expressing the conditions for the existence of a certain class of relative equilibria, some of which have a simple rotation, whereas others perform a double rotation, and he describes their qualitative behaviour. Table of Contents  Introduction
 Background and equations of motion
 Isometries and relative equilibria
 Criteria and qualitative behaviour
 Examples
 Conclusions
 Bibliography
