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I remember back in the day, I took a course in cosmology where I was taught that there are three possible types of universes based on their curvature, the flat universe which may be compact or infinite, the spherical which is compact and the hyperbolic which is infinite.

I don't understand why a hyperbolic universe must be infinite, I know many manifolds which admit a hyperbolic structure and are compact, in fact most manifolds are hyperbolic, an example is, all the surfaces of genus greater than or equal to 2.

Now I am not a physicist and don't have good physics intuition, so I want to ask why does hyperbolic universe means infinite universe? Have I misunderstood something?

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You're correct that in general a hyperbolic geometry does not imply the manifold is infinite: it could have some non-trivial topology and be compact (e.g. 3-torus). In FLRW cosmology we usually assume$^1$ the topology is simply-connected, in which case negative curvature does imply an infinite universe, but of course this isn't the only option.

e.g. see What are the allowed topologies for a FRW metric?

Edit:
$^1$I'd also add that if one picks some of these more interesting compact topologies with constant negative curvature for the FLRW metric, then there are no continuous group isometries (no global Killing vector fields) and the spacetime is not isotropic at every point. So there are physical reasons why people often make the assumption that hyperbolic implies infinite, but it should be stated explicitly.

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  • $\begingroup$ en.wikipedia.org/wiki/… This says homogeneity is necessary for this metric, and therefore no odd topology is acceptable. $\endgroup$
    – Buzz
    Jan 8, 2021 at 15:16
  • $\begingroup$ @Buzz what I mean is, starting from the FLRW metric, without assuming simply-connectedness, you can obtain non-trivial topologies by identifying different points on the spatial 3-manifold: but it does ruin the isotropy, which is the reason for considering the metric in the first place. So I agree if you want to stick to homogeneity and isotropy globally then you do restrict the topology, but you can consider deviations in general. $\endgroup$
    – Eletie
    Jan 8, 2021 at 15:40
  • $\begingroup$ @Buzz to be more specific, the symmetries implied by assuming isotropy and homogeneity (i.e. in FLRW spacetimes) determine the geometry locally, not globally. One is free to still make these identifications to get nontrivial spacetime topologies. I don't see anything on the wiki in disagreement with this. $\endgroup$
    – Eletie
    Jan 8, 2021 at 15:55
  • $\begingroup$ Hi @Eletie. I really am unable to understand this distinction. My understanding is that homogeneity implies, for example, that for any sufficiently large sphere of a specified volume in the universe, there is a value for its property, for example say density, which is very closely approximately the same as for any other such sphere. I would guess that the locality refers to the slight differences of the value in different spheres. A different topology would create much different observable values for different spheres. $\endgroup$
    – Buzz
    Jan 9, 2021 at 18:22
  • $\begingroup$ en.wikipedia.org/wiki/… says astrophysicists now agree that the universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime. $\endgroup$
    – Buzz
    Jan 9, 2021 at 18:27

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