  We consider the 3-body problem in 3-dimensional spaces of nonzero constant
Gaussian curvature and study the relationship between the masses of the
Lagrangian relative equilibria, which are orbits that form a rigidly rotating
equilateral triangle at all times. There are three classes of Lagrangian
relative equilibria in 3-dimensional spaces of constant nonzero curvature:
positive elliptic and positive elliptic-elliptic, on 3-spheres, and negative
elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative
equilibria exist only for equal values of the masses.
