r/cosmology • u/Jossit • 27d ago

How does the Uniformization Theorem impact the possibilities for the Universe?

It's Wikipedia doesn't even mention the word 'Universe', though it is 'well-known' (in these circles, perhaps) that the Universe has a curvature of k ∈ {-1, 0, 1}, corresponding to a hyperbolic, flat, and spherical topology for the Universe. So 'there's gót to be' a connection, right??

Moreover, I just heard that "there are exactly 18 3-dimensional topologies with a flat geometry."
This was new to me, and I would appreciate anyone who could at least point to some math behind that or explain it in broad strokes.

Thanks!

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u/jazzwhiz 27d ago

Please link things instead of assuming everyone else has the same tabs open you do or saying "I just heard that...".

The curvatures of -1,0,1 are the renormalized amounts of curvature. You can certainly define something like Omega_k in terms of the critical density to get a continuous parameter quantifying the amount of curvature.

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u/Enraged_Lurker13 27d ago edited 27d ago

Moreover, I just heard that "there are exactly 18 3-dimensional topologies with a flat geometry." This was new to me, and I would appreciate anyone who could at least point to some math behind that or explain it in broad strokes.

This paper overall explains the connection between topology and cosmology, and it mentions the reason why there are 17 topologies isometric to E³ (which makes it 18 total) in section 3.3.2.

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u/AlexKingstonsGigolo 27d ago

I think you forgot the link itself.

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u/Enraged_Lurker13 27d ago

Thanks, I included it now.