I Two-dimensional negative curvature space

[Jaime Rudas]

TL;DR

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?

 

[PAllen]

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.

 

[Ibix]

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 $k$.