Polygonal homographic orbits of the curved $n$-body problem

In the $2$-dimensional $n$-body problem, $n\ge 3$, in spaces of constant curvature, $\kappa \ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons. We further use this criterion to show that, for any $n\ge 3$, the regular $n$-gon is a polygonal homographic orbit if and only if all masses are equal. Then we prove the existence of relative equilibria of nonequal masses on the sphere of curvature $\kappa >0$ for $n=3$ in the case of scalene triangles. Such triangular relative equilibria occur only along fixed geodesics and are generated from fixed points of the sphere. Finally, through a classification of the isosceles case, we prove that not any choice of three masses can form a triangular relative equilibrium.