Compactness and index of ordinary central configurations for the curved n-body problem

Abstract

For the curved n-body problem, we show that the set of ordinary central configurations is away from most singular configurations in H^3, and away from a subset of singular configurations in 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 (3n-4)(n-1)!/2 ordinary central configurations for given n masses if all ordinary central configurations of these masses are non-degenerate.