  We consider the 4-body problem in spaces of constant curvature and study the
existence of spherical and hyperbolic rectangular solutions, i.e. equiangular
quadrilateral motions on spheres and hyperbolic spheres. We focus on relative
equilibria (orbits that maintain constant mutual distances) and rotopulsators
(configurations that rotate and change size, but preserve equiangularity). We
prove that when such orbits exist, they are necessarily spherical or hyperbolic
squares, i.e. equiangular equilateral quadrilaterals.
