We answer here a question posed by F. Diacu in 2012 that asked whether there exist relative equilibria on S 2 and H 2 that move in a plane non-perpendicular to the rotation axis. For 3-body non-geodesic ordinary central configurations on S 2 and H 2 , we find all relative equilibria that move in a plane perpendicular to the rotation axis. We also show that the set of shapes of 3-body non-geodesic ordinary central configurations on S 2 and H 2 is a 3-dimensional manifold. Then we conclude that

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... most all 3-body relative equilibria move in planes non-perpendicular to the rotation axis. 1. Introduction. The Newtonian N -body problem studies the dynamics of N particles moving according to Newton's laws of motion in R n , where n is 2 or 3. After the discovery of hyperbolic geometry in the 19th century, the idea that the universe could be a 3-sphere or hyperbolic 3-sphere occurred. Thus the dynamics of particles in H 3 and S 3 was considered. This problem also attracted attention later from the point of view of quantum mechanics [18] and the theory of integrable dynamical systems [15, 19] . Readers interested in its history can consult [3, 20] . The topic of relative equilibria was initially considered in the 2-dimensional case [14] . After the equations of motion were written in extrinsic coordinates in R 4 for S 3 , and in the Minkowski space R 3,1 for H 3 , the relative equilibria were classified and studied in detail (see [4] and its references). In this set up, the matrix Lie groups SO(4) and SO(3, 1) serve as symmetry groups, which make the study of the 3-dimensional relative equilibria easier. With this new approach, new results on relative equilibria could be obtained, [3, 2, 4, 10] , as well as on other topics, 2010 Mathematics Subject Classification. Primary: 70F15; Secondary: 70F07.