Talk:Nash embedding theorem

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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)

Contents 1 Local isometric embedding 2 The value of n 3 Erratum 4 Gauss formula

[edit] 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)

[edit] 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)

[edit] Erratum

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

[edit] 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)