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I got the idea that expansion of the universe is not to somewhere, it is just getting stretched of spacetime since a point of singularity. And I know that universe was calculated as flat (which means sum of angles of a triangle is exactly π ignoring local instrinsic curvatures caused by celestial masses) with a very small error rate. But still universe is possible to be hyperbolical or elliptical. If it is elliptical, when you travel on a straight route, you will eventually arrive at your starting point (we assume expansion of the universe is frozen). So, can we say that in that travel I am actually circumnavigating on the surface of a higher dimensional sphere and that means spacetime has an extrinsic curvature beside its intrinsic curvature.

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If there was a higher-dimensional space in which the universe were embedded, then yes. But there is no need for any such space: the 3-sphere can exist happily on its own in 3 dimensions.

If this seems weird, take the simpler example of a cylinder. We’re used to thinking of the cylinder as a 3-dimensional volume, but the cylinder is sufficiently defined in 2 dimensions thus: take a 2-dimensional semi-infinite plane and glue together one pair of opposing edges. This “gluing” operation can be imagined as a topological rule defined on the plane, and needs no higher dimension in order to be realized.

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  • $\begingroup$ I read somewhere that if universe is sphere, it has that much of radius etc. Is that meaningless. If not, how can we define its radius when we don't need more spatial dimensions. $\endgroup$ – Alper Apr 3 '18 at 14:16
  • $\begingroup$ Ok, understood. For your example, it doesn't implied existence of an actual cylinder with a circular cross-section. Just defining topology of the plane by using analogy of a cylinder. $\endgroup$ – Alper Apr 3 '18 at 14:30
  • $\begingroup$ In response to your first question: walk in one direction until you return to your starting point. Divide this distance by $2\pi$. This is the radius. $\endgroup$ – bapowell Apr 3 '18 at 21:28
  • $\begingroup$ Yes, if you sure that your five dimensional universe is a sphere, instead of an elliptoid. $\endgroup$ – Alper Apr 3 '18 at 21:34
  • $\begingroup$ The whole point is that it’s 4 dimensional: there is no need for the embedding space. $\endgroup$ – bapowell Apr 3 '18 at 23:16

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