Hilbert's theorem and the pseudosphere

[diegzumillo]

Hey there,

My current objective is to study dynamics on compact spaces of negative curvature. This requires me to learn at least some basics of topology and differential geometry, completely new subjects to me!

I'm trying to understand this article from Balazs and Voros "Chaos on the pseudosphere", close to the beginning it is stated that it's prohibited for a compact surface of constant negative surface to be embedded in a 3-dimensional Euclidean space, and they use this to motivate the study of the pseudosphere on the Minkowski 3-d space.

But, from what I've been reading about DG and Topology there is no such prohibition, instead, the Hilbert's theorem (which I think is being referred) states that it is impossible to embed a COMPLETE and compact surface of constant negative surface in any $$\Re^{3}$$, which includes Minkowski space.

The understanding I'm making of this is that the article is 'imprecise', the Minkowski space is simply a more convenient way of working with the pseudosphere because of simpler equations, but it's not a complete surface (the edges are a singularity, therefore it's not a complete surface) so there must be some equivalent surface in the euclidena 3-d space... does this makes sense? is the article correct and I'm making a huge mess here?

Thanks everyone :) I hope I'm posting this in the appropriate forum.

 

[diegzumillo]

A little update on my huge mess, I mean, question.

I was reading this article "Pseudospheres in geometry and physics: from Beltrami to De Sitter and beyond" from Bruno Bertotti, Roberto Catenacci and Claudio Dappiaggi (available on arxiv.org) and it confronts these topics directly! I'm still trying to figure it out, though.