All the Lagrangian relative equilibria of the curved 3-body problem have equal masses

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.