I Two-dimensional negative curvature space

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TL;DR Summary
Can a two-dimensional space have constant negative curvature?
In this PF Insight says:

(...) if you want a two-dimensional space to be homogeneous and isotropic, there are only three possibilities that fit the bill: space can be uniformly flat, it can have uniform positive curvature, or it can have uniform negative curvature.

Can a two-dimensional space really have constant negative curvature? That is, can a two-dimensional surface with uniform negative curvature be constructed?
 
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Jaime Rudas said:
TL;DR Summary: Can a two-dimensional space have constant negative curvature?

In this PF Insight says:



Can a two-dimensional space really have constant negative curvature? That is, can a two-dimensional surface with uniform negative curvature be constructed?
Yes, but it can’t be smoothly isometrically embedded in Euclidean 3 space.
 
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You can read it off from the FLRW metric with ##dt=d\theta=0##: $$ds^2=\frac{1}{1-kr^2}dr^2+r^2d\phi^2$$Run that through the usual machinery and you get the Kretschmann scalar ##R_{ijkl}R^{ijkl}=2k##, which is obviously everywhere constant positive, negative or zero depending on your choice of ##k##.
 
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https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
The formal paper is here. The Rutgers University news has published a story about an image being closely examined at their New Brunswick campus. Here is an excerpt: Computer modeling of the gravitational lens by Keeton and Eid showed that the four visible foreground galaxies causing the gravitational bending couldn’t explain the details of the five-image pattern. Only with the addition of a large, invisible mass, in this case, a dark matter halo, could the model match the observations...
Hi, I’m pretty new to cosmology and I’m trying to get my head around the Big Bang and the potential infinite extent of the universe as a whole. There’s lots of misleading info out there but this forum and a few others have helped me and I just wanted to check I have the right idea. The Big Bang was the creation of space and time. At this instant t=0 space was infinite in size but the scale factor was zero. I’m picturing it (hopefully correctly) like an excel spreadsheet with infinite...

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