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I'm trying to understand this snippet from Wikipedia, in particular the section I've emphasized:

The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite.[14] Many textbooks erroneously state that a flat universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe.[14] For example, Euclidean space is flat, simply connected, and infinite, but the torus is flat, multiply connected, finite, and compact.

https://en.wikipedia.org/wiki/Shape_of_the_universe#Curvature

So if the universe has flat curvature, it can be either infinite or bounded with a 4 dimensional shape (compact). But why can't it be simply connected, like a 4 dimensional sphere? That would seem to be the most obvious shape to me for a finite universe.

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    $\begingroup$ The sphere $S^n$ is never flat, except for $n=1$. $\endgroup$ – AccidentalFourierTransform Dec 6 '17 at 15:57
  • $\begingroup$ @AccidentalFourierTransform Flat means you can draw parallel lines on it and they neither converge no diverge, but remain parallel, right? But you can draw latitude and longitude lines on the earth, and they're parallel. $\endgroup$ – John Dec 6 '17 at 15:59
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    $\begingroup$ Um, are you trying to say that the earth is flat? $\endgroup$ – AccidentalFourierTransform Dec 6 '17 at 16:07
  • $\begingroup$ @AccidentalFourierTransform haha, good one. I guess I'm saying that I don't fully understand what "flat" means in the topological sense. $\endgroup$ – John Dec 6 '17 at 16:14
  • $\begingroup$ @John: Flat means you can draw parallel straight lines that neither converge nor diverge. The lines of constant longitude converge; the lines of constant latitude are not straight: except for at the equator, following a line of constant latitude (except the equator) requires that you are always turning off of the geodesic (great circle) tangent to your current motion. $\endgroup$ – RLH Dec 6 '17 at 16:15
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A flat space means you can draw parallel straight lines that neither converge nor diverge.

On a two-sphere, this idea shows up as the lines of constant longitude converging and the lines of constant latitude not being straight: following a line of constant latitude (except the equator) requires that you are always turning off of the geodesic (great circle) tangent to your current motion.

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  • $\begingroup$ For example, near the poles, longitudes are small circles, no where near straight lines. $\endgroup$ – PyRulez Feb 5 '18 at 23:03

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