# Shape of the universe: why does “sphere” (positive curvature) mean finite?

Good day.

Wikipedia article

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

states:

"For example, a universe with positive curvature is necessarily finite. Although it is usually assumed in the literature that a flat or negatively curved universe is infinite, this need not be the case if the topology is not the trivial one."

Let me propose a constuction of a spehical and infinite universe. Imagine a sphere $S_0$ with two points $s_0^a$ and $s_0^b$. Consider these points as branch points in the usual meaning (we have from complex manifolds). You need to go twice (in my construction) around (any of these points) to come to the same "place". For simplicity imagine a line $l_0$ connecting these points $l_0(s_0^a,s_0^b)$ which represents a cut. If you cross the cut you end up in a different spere $S_1$. Hopefully I am clear enough. The speres $S_0$ and $S_1$ share points $s_0^a$ and $s_0^b$ and you go from one sphere to the other by crossing the cut. Almost everywhere we have a spherical curvature (locally).

Chosing a diferent pair of points $s_{-1}^a$ and $s_{-1}^b$ one adds to $S_0$ a "left" neighbout $S_{-1}$. In a repetitive way one builds an infinite chain of spheres $\{S_i\}_{i=-\infty}^{i=\infty}$ which represents an infinite universe.

What is wrong with this construction? Let me think:

• Is a branch point in contradiction with mathematical apparatus describing univers? Well, if a black hole with infinite curvature is not, why a brach point should be? At branch point everything is finite. A discontinuity appears in a curvature: if in a neigbouring point the circle length is $2\pi r-\varepsilon$ than at the branchpoint it is $4\pi r-\varepsilon$ (well, derivative is not finite). Still, it seems to me "better" then an infinite curvature.

• Does branch points create an un-isotropic universe? Well, for a single sphere there might be "preferred" directions: those pointing to the branch points. But the universe as whole, i.e. the infinite chain of spheres may have these branch points distributes "randomly" so no-one can really talk about any specific direction in the universe as whole.

Sub-question: a flat but finite universe (mentioned at Wikipedia) seems very interesting to me. Yes, I could read the link given there, but I am very bad at this topic, I would not understand. I imagine it must be some kine of smart "tiling of a plane". Question: is there a simple way (example) how flat but finite universe can be explained?

• To answer the question in your last paragraph: consider the torus. This can be made flat in dimensions 4 and higher. See the Clifford Torus. By "4 and higher" I mean the dimension of the space it is embedded in: the Clifford torus itself is a two dimensional manifold. If you want to forget the embedding (and, I believe, cosmologists always want to because the Universe is not thought to be embedded in anything), the existence of the Clifford torus proves that we can give the product of two 1-circles a Riemannian structure with a flat metric. – WetSavannaAnimal Feb 24 '17 at 10:37
• I think there are two separate issues: "being flat" and "being universe". OK, a finite torus can be made flat (locally). But can it be made "universe"? By "universe" we understand somethink isotropic. Is torus isotropic? Without having deep understanding I would say no: there are (two) preferred directions: the one of the "shorter diameter" and the one of the "longer diameter" - rougly speaking. – F. Jatpil Feb 24 '17 at 10:48