Talk:Whitney embedding theorem

message: 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)
replies: text: Smoothly. (done) Melchoir 20:21, 4 June 2006 (UTC)
replies: text: Thanks. Kevin Lamoreau 19:21, 5 June 2006 (UTC)
replies: text: It's not true that "all isometric embeddings are smooth", since smooth means infinitely diffferentiable.  For example, a square curve, of side equal to π/2, parametrized by arclength in the plane is an isometric embedding of the unit circle, and this isometric embedding is not even once-differentiable.Daqu (talk) 16:47, 27 February 2009 (UTC)
replies: text: What are you responding to?  It looks like you maybe put your comment in the wrong place?  There's no isometric anything in the comment above yours.  This article is about the Whitney embedding theorem, which makes no comment about metrics. Rybu (talk) 03:00, 2 March 2009 (UTC)
replies: text: The first sentence of this section includes the clause "i.e., all isometric embeddings are smooth."Daqu (talk) 02:49, 31 March 2009 (UTC)
thread_title:  What type of embedding does the Whitney embedding theorem apply to? 

message: Hi Nbarth, one of your recent edits had a mistake in it.  The mobius band isn't the only non-orientable manifold that embeds in R^3, for example.  All compact connected non-orientable manifolds with non-empty boundary embed in R^3.  I also brought back the comment that the strong Whitney embedding theorem is best-possible in dimensions that are a power of 2.  Maybe it doesn't deserve to be in the front matter of the article but it should be somewhere.  Your revision had it deleted and replace with some redundant info on the weak immersion theorem. Redundant as in the sense that it was already mentioned elsewhere in the article.    Anyhow the spirit of your revisions seem okay to me modulo execution and one mistake.  Rybu (talk) 16:02, 6 May 2009 (UTC)
thread_title:  some recent edits 

message: That is, is it true that an infinite dimensional Banach manifold is embeddable in its corresponding Banach space? Are there known counterexamples?  — Preceding unsigned comment added by 193.188.47.46 (talk) 07:59, 23 October 2017 (UTC)
thread_title:  Can the theorem be extended to infinite dimensional Banach manifolds? 

message: The introduction to the article contains this passage:
"*The weak Whitney embedding theorem states that any continuous function from an -dimensional manifold to an -dimensional manifold may be approximated by a smooth embedding provided .  Whitney similarly proved that such a map could be approximated by an immersion provided . This last result is sometimes called the Whitney immersion theorem."
This has the inequalities exactly backwards. It is the dimension of the larger manifold (into which the embedding or immersion occurs) that must be on the higher side of the inequality (not the dimension of the manifold that is being embedded or immersed).
I hope someone knowledgeable about this subject will fix these serious errors. 2601:200:C082:2EA0:6D35:62ED:104D:48D0 (talk) 04:45, 4 March 2023 (UTC)
thread_title:  Errors