Let me preface this post by saying that I only have a very cursory understanding of general relativity.(adsbygoogle = window.adsbygoogle || []).push({});

I happen to know that if we assume the cosmological principle, then the hypersurface ##\Sigma_t## of the spacetime manifold ##M##, for any positive ##t##, is either a 3-sphere, a 3-hyperboloid or just flat 3-space depending on the value of ##k## in the FRW-metric. Now, the assumptions of homogeneity and isotropy are made very mathematically precise. However, we know that the universe is onlyapproximatelyhomogeneous and isotropic, so our mathematical notions of homogeneous and isotropic do not exactly reflect what we observe in the universe. What consequence does this have for the shape of ##\Sigma_t##. Will it only beapproximatelya 3-sphere, for example? Perhaps a 3-spheroid? Or is the shape completely different and/or indeterminable?

EDIT: If the answer is that indeed the shape isapproximatelya 3-sphere, for example, how do we go about proving this? I get that the universe is almost an FLRW spacetime, but that does not mean that its shape has to be as if it wereexactlyan FLRW spacetime, does it?

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# B Shape of the universe and the cosmological principle

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