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. —Preceding unsigned comment added by 72.95.242.63 (talk) 21:07, 9 October 2009 (UTC)

There is the article pseudo-Riemannian space which does mention the hyperbolic metric. It is not possible to give a complete analogy with Riemann space because the quadratic differential form does not define a metric in the same way.JFB80 (talk) 19:46, 12 September 2013 (UTC)

An Unusual Explanation Of Hyperbolic Space

Just happened to bump into this old Discover article (Knit Theory March 2006 issue; written by David Samuels; photography by Richard Barnes). Could be useful to somewhat analogically explain hyperbolic space. —Preceding unsigned comment added by Komitsuki (talk • contribs) 19:08, 17 May 2010 (UTC)

Does the hyperbolic model really have negative Gaussian curvature?

Everywhere you find stated that hyperbolic space has negative curvature with a saddle point structure. But is this actually true? Consider a hemispherical bowl. From the outside it is convex in all directions so its principal curvatures are positive and so the Gaussian curvature is also positive. But if you look at the bowl from the inside it is concave in all directions so principal curvatures are all negative giving again a positive Gaussian curvature as the product of two negative numbers. It is the same for the hyperbolic model. From the outside it has positive curvature in all directions but from the inside it has negative curvature in all directions. So it has positive Gaussian curvature however you look at it. Of course there is also the hyperbolic surface in one sheet which does have saddle point structure with negative Gaussian curvature but that is not a model of hyperbolic space.JFB80 (talk) 19:15, 12 September 2013 (UTC)

You are ignoring the fact that the hyperbolic model is embedded in Minkowski space rather than Euclidean space. That makes all the difference. JRSpriggs (talk) 22:52, 12 September 2013 (UTC)

Yes you are right and the article does say so. I had verified it some time ago by transforming to the hyperbolic analogue of spherical coordinates. What I don’t understand is how so many people (Helmholtz, Killing, Poincare etc) previous to Minkowski's introduction of his space in 1908 are said to have used the 'well known hyperbolic model' without using Minkowski space. I did look at Killing's article but saw nothing about it.JFB80 (talk) 05:55, 14 September 2013 (UTC)

I am not familiar with the history or those old papers, so whatever I say here is speculation. However, it would not surprise me at all if Minkowski's space was already well known before Minkowski. Mathematically it is very simple, and thus probably very old. Minkowski's innovation was probably merely to notice that it was applicable to the physical world described by special relativity. JRSpriggs (talk) 09:17, 14 September 2013 (UTC)

Goal of this Hyperbolic space page

Hi

I want to make this page the main page for higher dimensional (3 dimensional and higher) hyperbolic geometry. For me this is mostly to "unburden" the hyperbolic geometry page form the higher dimensional "stuff". Bits of the hyperbolic geometry page that still needs to be included here are the formulas for spheres and balls:

The text I removed from the hyperbolic geometry page was:

The surface area of a sphere is

$4\pi R^{2}\sinh ^{2}{\frac {r}{R}}\,.$

The volume of the enclosed ball is

$\pi R^{3}\sinh {\frac {2r}{R}}-2\pi R^{2}r\,.$

For the measure of an n-1 sphere in n dimensional space the corresponding expression is

$\Omega _{n}R^{n-1}\sinh ^{n-1}{\frac {r}{R}}\,$ where the full n-dimensional solid angle is

$\Omega _{n}={\frac {2\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}\right)}}\,$ and $\Gamma \,$ is the Gamma function.

The measure of the enclosed n ball is:

$\Omega _{n}R^{n-1}\int _{0}^{r}\sinh ^{n-1}{\frac {s}{R}}ds\,.$

See further Talk:Hyperbolic geometry#Removed spheres and balls sections

But i am wondering where to put it on this page, also some cross linking is needed.

can we work together to do this?WillemienH (talk) 11:18, 12 April 2015 (UTC)

Modifications and TODO list

I started editing this page as it seemed a bit bare-bones and not super clear about what it wanted to be about. I removed most of the details on the specific models as they have their own articles. Maybe some details of the isomorphisms between the various models should be added back.

There's plenty of stuff that could (should?) be added, for instance:

• hyperbolic isometries and the isometry group;
• hyperplanes and hyperbolic polyhedra;
• discrete subgroups of isometries (in the "hyperbolic manifold" section);
• the boundary at infinity and harmonic analysis;

There's also a lot of results on minimal surfaces and such which i am not familiar with, and probably a lot more i'm not even aware of. The article is also very much lacking in references, i'll add some (eg. to Ratcliffe's book or Milnor's article on "the first 150 years") when have the time to locate them precisely. jraimbau (talk) 16:39, 20 December 2021 (UTC)

Difference between hyperbolic space and hyperbolic geometry?

What is the difference between hyperbolic space and hyperbolic geometry? Reading the comments here, it seems that some people regards hyperbolic geometry to be a broader term than hyperbolic space. So can it be said that hyperbolic space is a form of hyperbolic geometry? And if so, what kinds of hyperbolic geometries are there that do not also count as hyperbolic spaces?

Also, some people seem to consider hyperbolic spaces to strictly be of dimension 3 or greater (i.e., not including the hyperbolic plane). Why? This seems like a rather arbitrary constraint to me. What would be the purpose of discriminating between different spaces in this way? To me, this only seems to add a layer of linguistic complexity that isn't really justified. —Kri (talk) 16:28, 4 January 2022 (UTC)

Hyperbolic geometry is the study of the geometric properties of hyperbolic space, like Euclidean geometry is the study of properties of Euclidean space. As to what hyperbolic space is: by default it should be used to denote the spaces described in this article, since this is the historical meaning of the term (since Klein), and as far as i know it is still the primary meaning today. One ambiguity is that a priori there is no reason for the curvature to be normalised to -1, scaling the metric results in a space with the same geometric properties (up to constants in metric equalities, etc.), this is done only for the sake of clarity, so that "n-hyperbolic space" refers only to one object rather than an infinite family; but of course every simply-connected manifold of constant negative curvature is as deserving of the name "hyperbolic space" in principle.

I think the reason some people wanted a distinction between 2- and higher-dimensional spaces is that the hyperbolic plane can be characterised by axioms on points, lines, circles, etc. in the same way as the Euclidean plane, so "hyperbolic geometry" is the study of this axiomatic geometry in the vein of ancient euclidean geometry. At last this seems to be the assumption under which part of the "hyperbolic geometry" article is written. I agree with you that this is nonsensical, there is no serious reason to distinguish between dimensions in this article, i think in the current version there is no such discrimination. (Another reason is that in colloquial meaning "space" refers to 3-dimensional space, and so "hyperbolic space" may in context refer to actual hyperbolic 3-space ; but i don't think it makes sense with respect to this article either). jraimbau (talk) 10:49, 5 January 2022 (UTC)

Hmmmmmm

Has anyone thought about HxExS geometry, or H3xS3? It just seems like they would be really cool for a game Also how would HxExS work? 92.9.33.11 (talk) 19:09, 9 February 2022 (UTC)

By H×E×S, do you mean one dimension each? H¹×E¹ is indistinguishable from E²; E²×S¹ is a flat 3-space that is periodic in one direction.

H³×S³ is a six-dimensional space; how would you use that in a game? —Tamfang (talk) 03:48, 27 November 2023 (UTC)

Hyperbolic space with other sectional curvature

It might be better to define "a hyperbolic space" in such a way that arbitrary constant negative sectional curvature is allowed, instead of trying to define "the hyperbolic space". After all, if someone comes across such a space they are still likely to call it "hyperbolic space", in the same way that a (hyper)sphere of radius other than 1 is still going to be called a "(hyper)sphere". If we tried to make a definition at sphere like "the sphere is the unique simply connected, 2-dimensional, orientable Riemannian manifold of constant sectional curvature equal to 1" I feel like there would be significant pushback. Emphasizing that hyperbolic space is "unique" (i.e. every hyperbolic space is isomorphic) doesn't seem like an essential part of the definition, but is worth discussing, as is currently done at § Hyperbolic manifolds. The comments there could plausibly be extended. –jacobolus (t) 01:18, 13 March 2024 (UTC)