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.

What different criticism can one make about this constuction?

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. Feb 24, 2017 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. Feb 24, 2017 at 10:48

1 Answer

I looked at the reference in Wikipedia that a positive curvature meant that it had to be finite. That reference was for Ellis and another person's paper, see at https://arxiv.org/abs/gr-qc/9812046

I didn't exhaustively search for that statement or discussion or treatment of the issue in the paper (really Lectures at a conference, very well written and exhaustive), which is over 80 pages, and did not find it. The paper treats many differential geometry options for the spacetime, but I did not find any topology. But I have seen the same claimed in multiple papers, lectures, books, etc, I have claimed it myself, just never seen it proved.

However, I understand that there is a theorem that says that any space in N dimensions that has constant positive curvature has the topology of a sphere. You have to have that positive curvature everywhere. If not, and you decide to make an exception for a point or a line or curve or volume, yes, I think you might be able to create those connected near-spheres. Still, you are not then positive and constant curvature everywhere. Note that the constant curvature is not a problem for cosmology, the cosmological solutions have constant (positive, negative or zero) curvatures. Note also that it assumes homogeneous and isotropic spaces. If you deviate from those I don't know the results.

You can have negative and zero constant curvatures and still construct non trivial topologies. Just not with positive constant curvature since it closes on itself.

And as for a black hole, it remains an issue but the singularity happens Ina regime where quantum gravity is applicable, and we still don't have a good theory of quantum gravity.

For flat,yes it can be finite due to the tiling, and maybe other ways. Flat spacetime can be infinite (and would be with a trivial topology), or finite. I think I saw an article, and (I know) I wrote an answer in PSE that there were something like 17 (may not be the right number) flat non-trivial flat topologies possible. Of course eve never seen any evidence that any of them exists, but have not been able to totally rule them out. Physically it is thought that unless there were some very extreme effects soon after the Big Bang the topology should be trivial.

• Is the big bang itself not an extreme event? just saying xD Apr 12, 2020 at 12:20