Talk:Hyperbolic space

Content

This strikes me as a dubious article for hyperbolic space, since it is largely a discussion of the hyperboloid model. Gene Ward Smith 08:25, 17 April 2006 (UTC)

Yeah, I noticed that. I think this article should be more about the general properties of hyperbolic space, mentioning only a model when it makes it particularly easy to see something. Also some general stuff about applications and so forth. --C S (Talk) 08:48, 17 April 2006 (UTC)

We could move some of the material to hyperboloid model. But if we only talk about general properties, what distinguishes this article from the one on hyperbolic geometry? A general discussion of models might make sense, on the grounds that a hyperbolic space, concretely, means some model. Gene Ward Smith 20:31, 17 April 2006 (UTC)

Sure, a general discussion would be good. I think that the audience of hyperbolic geometry (because of the popularity of the term in various books, articles, etc.) is meant to be quite wide, while the audience for hyperbolic space is meant to be for those with more mathematical training. That's what I've observed in watching these articles develop. I think also that some editors had the notion that the geometry article should cover the plane, and higher dimensions in general should be covered here. Unfortunately, strictly speaking, hyperbolic geometry should cover a lot more of the general stuff and applications than hyperbolic space, since the term refers to more than hyperbolic space itself. Hyperbolic space could cover, as you say, a general discussion of the standard models and various other representations, including say, combinatorial representations and its relevance to delta-hyperbolic spaces.

In any case, clearly a reorganization of some sort is needed, which takes into account these issues. BTW, your work on these hyperbolic geometry related pages is greatly appreciated and needed. --C S (Talk) 22:46, 17 April 2006 (UTC)

I've moved the symmetry section to hyperboloid model, changing notation to fit that introduced there. The original section is the one below. Gene Ward Smith 08:21, 19 April 2006 (UTC)

The hyperboloid model is perhaps the least known model of hyperbolic space. It has some advantages: it is easy to describe the group of isometries for all dimensions (just as SO(1,n)); it is rather easy to describe the geodesics (as any non-empty intersection between a planes passing the origin and the upper part of the hyperboloid).

It would be good to explain the relation between the hyperboloid model and the other models in some detail. For example, stereographic projection of the hyperboloid model gives the Poincaré disk model. These relations between the models should be given as explicit as possible. Perhaps as many models as possible should be included in the article?

I also would like to thank the authors up to now for their effort writing this article. Pierreback 23:18, 26 April 2006 (UTC)

Symmetry

The group O(n,1) is the Lie group of $(n+1)\times (n+1)$ real matrices that preserve the bilinear form

$\langle x,y\rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}.$

That is, O(n,1) is the group of isometries of Minkowski space R^n,1 fixing the origin. This group is sometimes called the (n+1)-dimensional Lorentz group. The subgroup which preserves the sign of x₀ (if $\langle x,x\rangle <0$ ) is called the orthochronous Lorentz group, denoted O⁺(n,1).

The action of O⁺(n,1) on R^n,1 restricts to an action on Hⁿ. This group clearly preserves the hyperbolic metric on Hⁿ. In fact, O⁺(n,1) is the full isometry group of Hⁿ. This isometry group has dimension n(n+1)/2, the maximal dimension of the isometry group of a Riemannian manifold. Therefore, hyperbolic space is said to be maximally symmetric. The group of orientation preserving isometries of Hⁿ is the group SO⁺(n,1), which is the identity component of the full Lorentz group.

The orientation preserving isometry group SO⁺(n,1) acts transitively and faithfully on Hⁿ. Which is to say that Hⁿ is a homogeneous space for the action of SO⁺(n,1). The isotropy group of the vector $(1,0,\ldots ,0)$ is a matrix of the form

${\begin{pmatrix}1&0&\ldots &0\\0&&&\\\vdots &&A&\\0&&&\\\end{pmatrix}}$

where A is a matrix in the rotation group SO(n); that is, A is an $n\times n$ orthogonal matrix with determinant +1. Hyperbolic space Hⁿ is therefore isomorphic to the quotient space SO⁺(n,1)/SO(n).

The bilinear form $\langle \,,\,\rangle$ is the Cartan-Killing form, the unique SO⁺(n,1)-invariant quadratic form on SO⁺(n,1).

Questions about hyperbolic 1-space

For all the talk about the hyperbolic plane and even hyperbolic [i]n[/i]-space of [i]n[/i] > 2, hyperbolic 1-space, the "maximally symmetric, simply connected [of course any connected 1-dimensional manifold, even the circle, is simply connected]," 1-dimensional "Riemannian manifold with constant sectional curvature −1," is not often talked about, and I am curious about it. What is it called? The hyperbolic line? I can tell that hyperbolas are not examples of hyperbolic-1 space. How many dimensions of Euclidean space does it take to isometrically embed hyperbolic 1-space? Or what I'm really looking for is how many dimensions of Euclidean space does it take for hyperbolic 1-space to be embedded in and be as much itself, if you know what I mean, as the circle is in the Euclidean plane and the [i]n[/i]-sphere is in Euclidean [i]n[/i]+1-space. That may be eqivilent in all cases to a manifold being able to be isometrically embedded in a certain space, but I'm not sure. The number of dimensions it takes to isometrically embed hyperbolic 1-space could shed some insight into the number of dimensions it takes to isometrically embed hyperbolic 2-space, which I believe has been narrowed down to 4 or 5 now but I'm not sure if it's been proven that it doesn't take more than 5. Any answers to these questions would be appreciated. Kevin Lamoreau 17:34, 29 May 2006 (UTC) [edited by the same Kevin Lamoreau 05:06, 4 June 2006 (UTC) ]

The hyperbolic line doesn't actually exist. The reason is that Riemannian geometry is trivial in dimension 1; meaning that all 1-dimensional Riemannian manifolds are locally isometric. Therefore, the curvature of everything is 0. Negative curvature spaces exists only in dimension two or higher. -- Fropuff 05:47, 4 June 2006 (UTC)

Thanks Fropuff. I now get why negative curvature doesn't exist (and why, if it did exist by the absence of the absolute value function in the measurement of curvature for curves, a curve or section thereof with non-zero curvature would have both positive and negative curvature of the same magnitude as Tomruen said the circle had in his reply in Talk:Hyperbolic geometry). Your reply was definately helpful, as was Tom Ruen's. Kevin Lamoreau 20:07, 5 June 2006 (UTC)

Also note that "of course any connected 1-dimensional manifold, even the circle, is simply connected" is false. For example, the circle is not simply connected. 68.100.203.44 18:02, 31 August 2006 (UTC)

Hyperbolic metric

There appears to be no proper article or section of an article on the hyperbolic Riemannian metric. Although its "integrated" version (the metric space distance function) typically seems to be given, it would also be highly relevant to give the infinitesimal version.