Compactness and Index of Ordinary Central Configurations for the Curved \(N\)-Body Problem

For the curved \(n\)-body problem, we show that the set of ordinary central configurations is away from singular configurations in \(\mathbb{H}^{3}\) with positive momentum of inertia, and away from a subset of singular configurations in \(\mathbb{S}^{3}\). We also show that each of the \(n!/2\) geodesic ordinary central configurations for \(n\) masses has Morse index \(n-2\). Then we get a direct corollary that there are at least \(\frac{(3n-4)(n-1)!}{2}\) ordinary central configurations for given \(n\) masses if all ordinary central configurations of these masses are nondegenerate.