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For the curved n-body problem, we show that the set of ordinary centralconfigurations is away from most singular configurations in H^3, and away froma subset of singular configurations in S^3. We also show that each of the n!/2geodesic ordinary central configurations for n masses has Morse index n-2. Thenwe get a direct corollary that there are at least (3n-4)(n-1)!/2 ordinarycentral configurations for given n masses if all ordinary centralconfigurations of these masses are non-degenerate.

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