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==''C''<sup>''k''</sup> embedding theorem==
The technical statement appearing in Nash's original paper is as follows: if ''M'' is a given ''m''-dimensional Riemannian manifold (analytic or of class ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞), then there exists a number ''n'' (with ''n'' ≤ ''m''(3''m''+11)/2 if ''M'' is a compact manifold, or ''n'' ≤ ''m''(''m''+1)(3''m''+11)/2 if ''M'' is a non-compact manifold) and an [[isometric embedding]] ƒ: ''M'' → '''R'''<sup>''n''</sup> (also analytic or of class ''C<sup>k</sup>'').<ref>{{Cite journal|last=Nash|first=John|date=1956-01|title=The Imbedding Problem for Riemannian Manifolds|url=https://www.jstor.org/stable/1969989?origin=crossref|journal=The Annals of Mathematics|volume=63|issue=1|pages=20|doi=10.2307/1969989}}</ref> That is ƒ is an [[embedding#Differential Topology|embedding]] of ''C<sup>k</sup>'' manifolds and for every point ''p'' of ''M'', the [[derivative]] dƒ<sub>''p''</sub> is a [[linear operator|linear map]] from the [[tangent space]] ''T<sub>p</sub>M'' to '''R'''<sup>''n''</sup> which is compatible with the given [[inner product space|inner product]] on ''T<sub>p</sub>M'' and the standard [[scalar product|dot product]] of '''R'''<sup>''n''</sup> in the following sense:
: <math>\langle u,v \rangle = df_p(u)\cdot df_p(v)</math>
for all vectors ''u'', ''v'' in ''T<sub>p</sub>M''. This is an undetermined system of [[partial differential equation]]s (PDEs).
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The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into '''R'''<sup>''n''</sup>. A local embedding theorem is much simpler and can be proved using the [[implicit function theorem]] of advanced calculus in a [[Manifold#Charts|coordinate neighborhood]] of the manifold. The proof of the global embedding theorem relies on Nash's far-reaching generalization of the implicit function theorem, the [[Nash–Moser theorem]] and Newton's method with postconditioning. The basic idea of Nash's solution of the embedding problem is the use of [[Newton's method]] to prove the existence of a solution to the above system of PDEs. The standard Newton's method fails to converge when applied to the system; Nash uses smoothing operators defined by [[convolution]] to make the Newton iteration converge: this is Newton's method with postconditioning. The fact that this technique furnishes a solution is in itself an [[existence theorem]] and of independent interest. There is also an older method called [[Kantorovich theorem|Kantorovich iteration]] that uses Newton's method directly (without the introduction of smoothing operators).
==References==
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