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Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move on $M$ freely without frictions and mutually repulse from each other. When these $k$ particles stop at $(x_1,\cdots, x_k)$ and be stable under small disturbance, we just call $(x_1,\cdots,x_k)$ a "electron configuration". The collection of all "electron configuration"s form the "electron configuration space".

Question: Are there any reference about the "electron configuration space"? What formal names of these spaces should I search online?

Question: Any references about the cohomology ring of the "electron configuration space"?

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    $\begingroup$ On a sphere see en.wikipedia.org/wiki/Thomson_problem. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 17, 2015 at 2:11
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    $\begingroup$ I asked this question a long time ago, for both discrete and continuous charge distributions: mathoverflow.net/q/80731/12310 (I originally voted to close as a result, but our questions do seem to differ in the sense that I care about large k where you care about any k). $\endgroup$
    – Chris Gerig
    Commented Sep 17, 2015 at 3:18
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    $\begingroup$ Given a fixed $k$, is the solution for stable electron configurations unique up to an isometry of $M$? $\endgroup$
    – Shi Q.
    Commented Sep 17, 2015 at 7:42
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    $\begingroup$ Nope, it's generally far from unique, even on spheres. The first case seems to be 16 points on $S^2$, for which there are (at least) two local optima. As the number of points grows, the number of local minima seems to increase exponentially, but no proof is known. In higher dimensions there are cases with arbitrarily large numbers of non-isometric global minima (the configurations in the last line of Table 1 in arxiv.org/abs/math/0607446). $\endgroup$
    – Henry Cohn
    Commented Sep 17, 2015 at 14:37
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    $\begingroup$ Another issue you might want to clarify is the dimensionality $n$ of M. If $n\ne3$, then retaining the form of Gauss's law is not consistent with a $-2$ exponent of the Coulomb force law (at short distances -- it doesn't make sense to talk about such a force law at long distances in a curved space). And are you imagining M as having only intrinsic structure, or as being embedded in a higher-dimensional space? If the latter, then Earnshaw's theorem doesn't apply. $\endgroup$
    – user21349
    Commented Sep 17, 2015 at 15:24

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There seems to be a considerable body of work on this:

Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds

D.P. Hardin, , E.B. Saff1,

More by Hardin and Saff.

http://personales.unican.es/beltranc/archivos/FoCMBeltran2011volume.pdf (Refers to work by Noam Elkies, who will probably have a much more in depth response)

Perhaps most relevant: Papers by Burton Randol and coworkers:

Burton Randol. Stable configurations of repelling points on manifolds. Proc. Amer. Math. Soc., 142:2769–2773, 2014.

Nechaeva and Randol

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