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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 ''n'' ≤ ''m''(''m''+1), or (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=January 1956|title=The Imbedding Problem for Riemannian Manifolds|url=https://www.jstor.org/stable/1969989|journal=The Annals of Mathematics|volume=63|issue=1|pages=20|doi=10.2307/1969989|jstor=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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