Stability of Fixed Points and Associated Relative Equilibria of the 3-Body Problem on \({\mathbb {S}}^1\) and \({\mathbb {S}}^2\)

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to \({\mathbb {S}}^1\), but unstable if the bodies are considered in \({\mathbb {S}}^2\).