There are mainly three theories about the shape of the universe: it can have a shape with zero curvature (flat), positive curvature (spherical) or negative curvature (hyperbolic). My question is "How can we prove that a non-euclidean shape is infinite or finite?" Mathematically, is there any way to prove that a hyperbolic shape is infinite ? Should I use limits or something else?
2026-03-26 06:29:58.1774506598
How to prove that a non-euclidean shape is infinite or finite?
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Mathematically one usually studies manifolds of constant curvature. These properties pretty much define the manifold, so everything else can be concluded from that. But the real universe definitely doesn't have constant curvature, as masses cause additional spacetime curvature. And while the concept of manifold entails that you'll never “hit a wall”, there is nothing to say that this holds for the real universe.
By analogy, you can't distinguish a sphere from a hemisphere until you see the boundary, or have explored half the sphere without finding one.