Compactification and the metric tensor

Question

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.

Answer 1

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