In this article we characterize the existence of ring solutions to the n-body problem on \(\mathbb {S}^2\) , parametrized by the circumradius \(a \in (0,1)\) . For configurations with identical masses \(m_j= m, j=1, \ldots , n\) , we show that when n is even, there exists a single bifurcation value associated with \(\omega ^2/m\) (where \(\omega \) is the angular velocity), for which there may exist zero, one, or two admissible values of a for which ring solutions exist, while for odd n, there are two bifurcation values of \(\omega ^2/m\) , leading again to zero, one, or two possible values of a for which ring solutions exist. On the other hand, we prove that if \(\frac{n}{2}\) or \(\frac{n-1}{2}\) is even, then there exists a value of the circumradius \(a_* \in (0,1)\) such that the ring solution admits non-identical masses. In this case, we also provide an explicit parameterization of the corresponding mass distribution. Important differences with the Newtonian case are found.
