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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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It is impossible to be simultaneously spherical and flat, unless you refer to being locally flat. Spheres are a surface with positive curvature by necessity. As it is a curved surface.

So, no, it is not possible to be a spherical Minkowski Space. Solvable by basic geometry.

But, if we do live on a De-Sitter Space (Positive Curvature) it would appear topologically flat except at very large distances. Think a very small ant walking on the surface of a large ball, the surface they walk on has such a small curvature on those scales that it appears flat. Scale it up, and this would apply to the Earth as well, and scale it up more, it would apply to the universe also.

And you mentioned lines of latitude and longitude, these are not actually straight lines, regardless of how they appear on a Mercator projection. They are actually curved lines set across it. As one can see here: https://c.tadst.com/gfx/1200x630/longitude-and-latitude-simple.png?1

And if one does mark straight lines, they would be Great Circles, and all Great Circles on a spherical surface do intersect. Therefore true parallel lines are an impossibility on a spherical surface.

There is this statement on Wikipedia, which, while not the best source, does serve to back up my point.

In spherical geometry, all geodesics are great circles. Great circles divide the sphere in two equal hemispheres and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called parallels of latitude analogous to the latitude lines on a globe. Parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of the sphere.

And if lines must curve to not intersect, they are not parallel, as parallel lines are necessarily straight.

I apologize if this sounds arrogant or rude. That is not my intention.

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