Talk:Whitney embedding theorem

From Wikipedia, the free encyclopedia

I assume that there are at least two versions, with different conditions, yielding 2n and 2n+1 as the dimension. Can anyone authoratively clarify this? --Pjacobi 01:46, August 27, 2005 (UTC)

2n is stronger, but often it is proved only for 2n+1 sinse it is much easier. Tosha 01:04, 23 September 2005 (UTC)

Are the hyperbolic plane and hyperbolic n-space smooth and second countable? In other words, are such manifolds (the hyperbolic plane being a surface, or 2-manifold) covered by this theorem? Kevin Lamoreau 04:20, 7 November 2005 (UTC)

Technically, yes. But the smooth structure of hyperbolic n-space is identical to that of Euclidean n-space, so the Whitney theorem is unnecessary; you can already smoothly embed Hⁿ in Rⁿ, let alone R²ⁿ. This embedding is going to screw up the curvature, but curvature isn't a property of smooth manifolds, so we're not worried about that here. Melchoir 06:14, 24 February 2006 (UTC)

Thanks for your reply, Melchoir. What I am actually interested in is knowing what dimensions of Euclidean space hyberbolic n-spaces of various n (particularly hyperbolic 1-, 2- and 3-space) are "naturally" contained in. In other words, if I were to take the hyperboloid model of the hyperbolic plane or the equivilent model for hyperbolic 1- or 3-space and converted the graph from the Minkowski space R^1,n (n being the dimension of the hyperbolic space in question) to the infinate-dimensional Euclidean space R^∞, first, could I make such a conversion without altering the graph beyond the fact that I'm graphing it in a different space, second, would the resulting graph lie entirely within a "hyperplane" (although with an infinite codimension) of Euclidean m-space of finite m and third, could anyone tell me what m would be for n (the dimension of the hyperbolic space) = 1, 2 and 3. I thought at first that saying a manifold could be embedded in a cetain space was equivalent to saying that it could be graphed in that space and be as much "itself," if you know what I mean, as a sphere is in Euclidean 3-space or as any n-sphere is in Euclidean (n + 1)-space. Perhaps that is the case for an isometric embedding (not merely a smooth embedding), but perhaps not. If that's not the case is there any term for (or always equivilent to) a manifold being able to be so contained in a certain space. In any case that's what I'm really interested in. I know it may not matter much in terms of any mathematical application, but I just want to know what dimension of Euclidean space hyperbolic 1-, 2- and 3-space can be "naturally" contained in. Kevin Lamoreau 05:16, 1 May 2006 (UTC) - edited by myself, Kevin Lamoreau 20:17, 1 May 2006 (UTC)

It does sound like you're asking about isometric embeddings. Well, 1-dimensional flat and hyperbolic space are geometrically and in every way the same. But at 2 dimensions and greater, my intuition says that it's impossible to isometrically embed H^n in Euclidean space of any dimension; it would be hard to keep the image from circling around and hitting itself. I'm sure there are theorems to this effect; you may get a more enlightened response at Wikipedia:Reference desk/Mathematics. Melchoir 20:26, 4 June 2006 (UTC)

The 2-dimensional hyperbolic plane admits an isometric embedding into 6-dimensional Euclidean space. See: Blanusa (Monatshefte Math. 59 (1955) 217-229) More generally, any compact Riemann manifold admits *some* isometric embedding into some (very high dimensional) Euclidean space. This is called the Nash Embedding Theorem and it is a "big deal". Anyhow, for the embedding of the 2-dimensional hyperbolic plane in R^6, here is an explicit formula from a sci.math.research post of Dave Rusin's:

Now I will describe the embedding of R^2 into R^6. This is not exactly what one would call an "obvious" construction, but there are certain patterns to it which suggest how Blanusa might have been led to it. We will send the point (u,v) to a point with six coordinates (x1, ..., x6) which are functions of u and v of the special forms

x1 = x1(u) x2 = f1(u) sin( v psi1(u) ) x3 = f1(u) cos( v psi1(u) ) x4 = f2(u) sin( v psi2(u) ) x5 = f2(u) cos( v psi2(u) ) x6 = v

I will describe the functions x1, f1, f2, psi1, psi2 (of one variable each) in stages.

Let [x] denote the integer part of x.

The functions psi1 and psi2 are periodic functions of |u| (period = 2) which on [0,2] are exponentials of linear maps: psi1(u) = exp( 2*[ (|u|+1)/2 ] + 5 ) psi2(u) = exp( 2*[ ( |u| )/2 ] + 6 ) There are discontinuities in the psi_i at certain integers but other parts of the construction will keep the x_i smooth.

Define two functions phi_i via certain normalized antiderivatives: writing F(x) = sin( pi x )/exp( sin^{-2}(pi x) ) we have phi1(u) = { (1/A) integral( F(x), x=0 to x= u+1 ) }^(1/2) phi2(u) = { (1/A) integral( F(x), x=0 to x= u ) }^(1/2) where A = integral( F(x), x=0 to x=1 ) = 0.141327... These functions phi_i are non-negative, periodic, and satisfy phi1^2 + phi2^2 = 1 and phi1(u) = phi2(u+1). You can think of the phi_i as being very smooth versions of |sin(pi u)| and |cos(pi u)|.

Now set f_i(u) = sinh(u) phi_i(u)/psi_i(u) for i=1,2 and define x1 to be an antiderivative of 1-(f1')^2-(f2')^2 having x1(0)=0.

Loosely speaking the mapping x1 sends lines in R^2 far enough away, and the the coordinates x2, x3, x4, x5 allow the points in these line to spin around in four perpendicular directions, with enough spinning to account for the fact that the images of lines are supposed to grow very long. The last coordinate x6 merely adds a motion in another perpendicular direction to separate points so that these curves don't self-intersect.

The metric which R^2 inherits from this embedding into R^6 comes out to ds^2 = du^2 + cosh(u)^2 dv^2, from which one finds the curvature to be constant and negative, making R^2 into the hyperbolic plane.

dave

[edit] Assumptions

This article, and Whitney immersion theorem, assumes a "second-countable... manifold". As explained at Topological_manifold#Technical_details, the word "manifold" implies second-countable. I'll remove the second-countable bit. Melchoir 06:04, 24 February 2006 (UTC)

It is "usually required" but not always required!, it is better to revert it Tosha 21:38, 4 March 2006 (UTC)

I think that by "usually required," the author of that section meant that most mathematicians have being Hausdorff and second-countable as a criteria for a topological space to be considered a topological manifold. If every manifold is (at least) a topological manifold, than most mathematicians may indeed believe that the word manifold implies second-countability. But I'm not sure that are not any non-topological manifolds (differential manifolds are topological, but people may sometimes speak of topological manifolds as those which are not differential), though. I just thought I'd add what I just did to try to acheive greater clarity. Kevin Lamoreau 19:05, 22 April 2006 (UTC)

On sentence of the Manifold#Topological_manifolds, reads, "It is customary to require that the space be Hausdorff and second countable [to be deemed a topological manifold]." Also, the section Manifold#Mathematical_definition begins by saying, "In topology, an n-manifold is a second countable Hausdorff space in which every point has a neighborhood homeomorphic to an open Euclidean n-ball, :" So it seems pretty clear that the general consensus of mathemeticians is that all manifolds are second-countable. I hope that information helps. Kevin Lamoreau 19:21, 22 April 2006 (UTC)

That is all true but it is also ok to leave it in the formulation (just in case), we could put it in "()".--Tosha 22:02, 22 April 2006 (UTC)

Second-countable is very commonly assumed for manifolds. The digression in the intro was distracting from the real content so I removed most of it. - Gauge 04:06, 28 June 2006 (UTC)

[edit] What type of embedding does the Whitney embedding theorem apply to?

Inspired by a realisation that there is more than one type of embedding (with definitions that are not equivilant, although satisfaction of some of those definitions imply satisfaction of others, i.e. all isometric embeddings are smooth), I thought I'd ask which type of embedding, as defined in the article of that name in Wikipedia, the Whitney embedding theorem applies to. In other words, in what way does the Whitney embedding theorem state that any smooth, second-countable (if that's not redundant) m-dimensional manifold can be embedded in Euclidean 2m-space? That information ought to be added to the first sentence of this article, as well as the defining sentences of all articles on embedding theorems in Wikipedia. It would only take the addition of one adverb for this article, although if the answer is that the Whitney Embedding Theorem only guarantees the existance of a smooth embedding (as opposed to an isometric embedding) of a smooth, second-countable m-dimensional manifold in Euclidean 2m-space, that probably ought to be specified somewhere in the article. I'd be happy to do make these editions myself, as I consider myself pretty good at coming up with encyclopidia-appropriate language when I know what I want to be (I don't care that much when I'm asking for information in discussion pages), but if someone else wants to go ahead and do that him/herself that's fin with me. Kevin Lamoreau 20:41, 1 May 2006 (UTC)

Smoothly. (done) Melchoir 20:21, 4 June 2006 (UTC)

Thanks. Kevin Lamoreau 19:21, 5 June 2006 (UTC)

Technically, the strong Whitney embedding theorem (SWET) only applies to connected 2nd countable Hausdorff manifolds. If you allow disconnected manifolds, then consider the two point space (as a 0-dimensional manifold) and find an embedding into R^0.

The proof of the SWET is done in several steps. You need the weak embedding theorem (embeds in 2n+1), then you consider n-dimensional manifolds with n>2, because this gives 2n>4 and so you can approximate a map from a 2-disc to R^{2n} by an embedding (by the weak embedding theorem). But just having the disc is not enough. You can not remove a single double point! Whitney went to a lot of trouble in his paper to show how you have to artifically introduce extra "local" double points so that you can use it to cancel with the "natural" ones. You handle 0, 1 and 2-dimensional manifolds seperately.

Anyhow, this article needs a lot of cleaning up.

[edit] connected (?)

Q: Any reason for the connected hypothesis??

A: Consider the case where the manifold is 0-dimensional. A 2-point set does not embed in a 1-point set.