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Cosmological measurements suggest that we live in a flat universe. However, what might be less clear is its topology. So could the flat universe have the form of a $T^3$-torus, i.e. the torus whose surface would be (diffeomorph to) 3D ?

For instance the Robertson-Walker metric for $k=0$ could turn out as a $T^3$ if certain point identification would apply.

Or would that contradict the rules of homogeneity and isotropy of the universe which is the basis for the Robertson-Walker metric ? If the latter were not the case, why would it be not possible ?

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    $\begingroup$ My understanding is the universe could conceivably have about any shape, provided it is big enough that anything we observe in our local neighborhood is flat within our current measurements. Homogeneity and isotropy are assumptions, not rules the real universe must follow. $\endgroup$
    – RC_23
    Commented Mar 16 at 16:39
  • $\begingroup$ Moreover, homogeneity and isotropy are assumptions that get tested from time to time, and have yet to be disproven. $\endgroup$
    – TimWescott
    Commented Mar 16 at 17:03
  • $\begingroup$ About the global properties of the Universe (i. its topology), the Wikipedia page could be usefull: en.wikipedia.org/wiki/Shape_of_the_universe $\endgroup$
    – Cham
    Commented Mar 16 at 17:36
  • $\begingroup$ @Cham Thank you for the useful hint. $\endgroup$
    – Frederic Thomas
    Commented Mar 16 at 23:15

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A universe with toroidal spatial surfaces is not in obvious conflict with general relativity, which is a local theory. Nor is it in conflict with the FLRW solution, which is separately applicable to different patches of the universe and says nothing about the universe as a whole.

It would likely be at odds with observations, however, if the length scale is smaller than the size of the observable universe. It should make repeating patterns in the cosmic microwave background and the large-scale structure of the universe, which would show up very clearly in the frequency-space analyses that we normally use to characterize these systems.

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