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

Rectangular orbits of the curved 4-body problem