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.

中文翻译：

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曲面$$N$$-体问题的普通中心构型的紧致性和指数

对于弯曲的\(n\) -body 问题，我们表明普通中心配置集远离\(\mathbb{H}^{3}\) 中具有正惯性动量的奇异配置，并且远离 a \(\mathbb{S}^{3}\)中奇异配置的子集。我们还表明，\(n\)质量的每个\(n!/2\)测地线普通中心配置都有莫尔斯指数\(n-2\)。然后我们得到一个直接的推论，如果所有的普通中心构型都是给定的\(n\)质量，那么至少存在\(\frac{(3n-4)(n-1)!}{2}\)普通中心构型这些团块是非退化的。