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.