If we have a manifold $M$ with a metric tensor $g_{\mu\nu}$ and want to compactify it, let's say by the circle $S^1$, how can we find the metric tensor for the compactified space?
Note: I am no expert in this topic, in fact I am just getting started.
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$\begingroup$ Might Mathematics be better suited for this question? $\endgroup$– Kyle KanosOct 15, 2017 at 20:09
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$\begingroup$ Have a look at Kaluza Klein theory for a toy model on how compactification works. $\endgroup$– JPhyOct 16, 2017 at 20:43
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1 Answer
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Typically one already has an ansatz for the compactified space metric, or one leaves it in the form $g_{\mu \nu}$. There are some restrictions though, such as requiring the Ricci cuvature of the metric to vanish if you started with a higher dimensional space of vanishing curvature and reduced on $S^1$. This reference should help out: https://arxiv.org/abs/1310.6353
