Talk:Nash embedding theorems

Untitled Discussion

What about the case k = 2?

Crust 1 July 2005 17:30 (UTC)

Does anyone know if the theorem holds for pseudo Riemanian manifolds also? The Infidel 10:37, 15 January 2006 (UTC)

Yes, it follows from original Nash's theorem (but the embedding is in pseudo-Euclidean space) sinse pseudo-Riem. metric is difference of two Riemaninnian metrics, but there is a proof which is much simpler and does not use Nash, it is mentioned in Gromov's book on partial differential relations. Tosha 14:02, 17 January 2006 (UTC)

Thank you. Do we have the original Nash theorem somewhere in wikipedia? I guess the idea is to decompose the matrix of the pseudo-metric as a difference of two positive-definite matrices and to look at the proof to verify that the embeding conserves both of them? The Infidel 09:54, 21 January 2006 (UTC)

Local isometric embedding

The section "C^k embedding theorem" includes the passage:

"The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into Rⁿ. A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus."

I don't doubt that the local embedding theorem is simpler than the global one, but I doubt that that it's as simple as this passage makes it sound. Can whoever wrote that please mention at least what the conclusion of the local embedding theorem is? That is: What is the lowest dimension d(n) such that every C^k Riemannian metric (k ≥ 2) on an open n-ball can be isometrically embedded in R^d(n) ? And is it fair to presume that the implicit function theorem here refers to an infinite-dimensional setting? Perhaps d(n) even depends on the choice of k ≥ 2 ??? Is it the same d(n) for a C^ω metric ??? Daqu 05:44, 10 January 2007 (UTC)

Burstin-Janet-Cartan Theorem

You can find it in Spivak's A Comprehensive Introduction To Differential Geometry, Vol 5, Chapter 11. --58.244.62.190 (talk) 18:05, 20 February 2010 (UTC)

The value of n

Can anyone explain why the value of n is sufficient when n=m²+5m+3? In the original paper this value is n=m(3m+11)/2, which is considerably bigger than m²+5m+3 for big m. Temur 04:58, 9 July 2007 (UTC)

I agree. Neither could I find any reference that mentions n=m²+5m+3 as a tighter upper bound. I am changing it to n=m(3m+11)/2 for compact manifolds, and n=(m+1)(3m+11)/2 for non-compact manifolds, as mentioned in Nash's original paper. If anybody can show a reliable reference with the tighter upper bound, please feel free to revert, but please add the reference. - Subh83 (talk | contribs) 19:05, 17 March 2011 (UTC)

The Wikipedia write the conditions in C^k embedding theorem are "(with n ≤ m(3m+11)/2 if M is a compact manifold, or n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) "，I think the correct conditions are "with n ≥ m(3m+11)/2 if M is a compact manifold, or n ≥ m(m+1)(3m+11)/2 if M is a non-compact manifold)". For example, if n=m-1 then the manifold M with dimension m will isometric to f(M)$\subset$ R^m-1, this seems impossible to me. — Preceding unsigned comment added by Wttwcl (talk • contribs) 01:16, 11 December 2013 (UTC)

No. The Nash C^k embedding theorem gives the value of N for which the manifold can surely be embedded isometrically in R^N. Of course that implies that for every n' > N isometric embedding will still be possible in R^n' (this is trivial). But it is also possible that M can be isometrically embedded in an Euclidean space with dimension n < N. The theorem is silent about that possibility (since that possibility will depend on the properties of the particular manifold (M, g)), and simply states that M can be embedded in an Euclidean space of dimension ≤ N.

I am not sure what you meant in your example though. I can't possibly see how a m-dimensional manifold can be isometrically embedded in a m-1 dimensional Euclidean space. - Subh83 (talk | contribs) 18:34, 11 December 2013 (UTC)

Thanks I got it, I thought the value m(3m+11)/2 and m(m+1)(3m+11)/2 are lower bounds for the embedding dimension like Whitey embedding theorem, which is NOT.

Hilbert's theorem

The article now claims that the theorem contradicts Hilbert's theorem. This seems very supprising to say the least since theorems aren't supposed to contradict each other. Anyone who can correct the article? —Preceding unsigned comment added by 65.93.150.157 (talk) 17:48, 12 November 2008 (UTC)

Erratum

Does this erratum imply that information in this article needs to be revised in any way? Earthsound 20:16, 16 October 2007 (UTC)

I guess this is the correct link to the erratum: http://web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/Erratum.txt. I am not sure if the conversation gives a corrected value for non-compact manifolds though. Was there a later publication with a correct value? - Subh83 (talk | contribs) 18:54, 11 December 2013 (UTC)

Added a sentence about this. - Subh83 (talk | contribs) 19:12, 11 December 2013 (UTC)

Gauss formula

Does the reference to the Gauss formula really refer to the Gauss-Bonnet formula? Jjauregui (talk) 15:59, 15 February 2008 (UTC)

NO--Tosha (talk) 07:12, 19 February 2008 (UTC)

Are you sure Nash did the analytic case in 1966? tk —Preceding unsigned comment added by 217.255.236.253 (talk) 01:30, 10 June 2008 (UTC)

it doesn't say this in the bible —Preceding unsigned comment added by 129.98.192.100 (talk) 02:53, 26 September 2010 (UTC)

Crumpled?

In the case of embedding in an ε-sphere, could "C¹ isometrically embedded" be regarded as loosely equivalent to the common idea of being "crumpled", like a crunched-up ball of stiff paper? -- The Anome (talk) 18:30, 14 May 2013 (UTC)

Adding Inline Citations

I just corrected the precise statement of the embedding theorem for C^k manifolds, and I wanted to add the citation inline, but doing this messed up how the references section looks. Can someone help? Light rays from the sun (talk) 06:17, 6 September 2021 (UTC)