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The '''Nash embedding theorems''' (or '''imbedding theorems'''), named after [[John Forbes Nash]], state that every [[Riemannian manifold]] can be isometrically [[embedding|embedded]] into some [[Euclidean space]].  Isometric means preserving the length of every [[rectifiable path|path]].  For instance, bending without stretching or tearing a page of paper gives an [[isometric embedding]] of the page into Euclidean space because curves drawn on the page retain the same [[arclength]] however the page is bent.

The first theorem is for [[continuously differentiable]] (''C''<sup>1</sup>) embeddings and the second for [[analytic function|analytic]] embeddings or embeddings that are [[smooth function|smooth]] of class ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other; the first one has a very simple proof and leads to some very counterintuitive conclusions, while the proof of the second one is very technical but the result is not that surprising.

The ''C''<sup>1</sup> theorem was published in 1954, the ''C<sup>k</sup>''-theorem in 1956.  The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by {{harvtxt|Greene|Jacobowitz|1971}}. (A local version of this result was proved by [[Élie Cartan]] and [[Maurice Janet]] in the 1920s.)  In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates.  Nash's proof of the ''C<sup>k</sup>''- case was later extrapolated into the [[h-principle]] and [[Nash–Moser theorem|Nash–Moser implicit function theorem]].  A simplified proof of the second Nash embedding theorem was obtained by {{harvtxt|Günther|1989}} who reduced the set of nonlinear [[partial differential equation]]s to an elliptic system, to which the [[contraction mapping theorem]] could be applied.

==Nash–Kuiper theorem (''C''<sup>1</sup> embedding theorem) {{anchor|Nash–Kuiper theorem}}==
'''Theorem.''' Let (''M'',''g'') be a Riemannian manifold and ƒ: ''M<sup>m</sup>'' → '''R'''<sup>''n''</sup> a [[short map|short]] ''C''<sup>∞</sup>-embedding (or [[Immersion (mathematics)|immersion]]) into Euclidean space '''R'''<sup>''n''</sup>, where ''n'' ≥ ''m''+1. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒ<sub>ε</sub>: ''M<sup>m</sup>'' → '''R'''<sup>''n''</sup> which is 
:(i) in class ''C''<sup>1</sup>,
:(ii) isometric: for any two vectors ''v'',''w''&nbsp;&isin;&nbsp;''T<sub>x</sub>''(''M'') in the [[tangent space]] at ''x'' &isin; ''M'',
:::<math>g(v,w)=\langle df_\epsilon(v),df_\epsilon(w)\rangle</math>,
:(iii) &epsilon;-close to ƒ:
:::<math> | f(x) - f_\epsilon (x) | < \epsilon ~\forall~ x\in M</math>.

In particular, as follows from the [[Whitney embedding theorem]], any ''m''-dimensional Riemannian manifold admits an isometric ''C''<sup>1</sup>-embedding into an ''arbitrarily small neighborhood'' in 2''m''-dimensional Euclidean space.

The theorem was originally proved by John Nash with the condition ''n'' ≥ ''m''+2 instead of ''n'' ≥ ''m''+1 and generalized by [[Nicolaas Kuiper]], by a relatively easy trick.

The theorem has many counterintuitive implications.  For example, it follows that any closed oriented Riemannian surface can be ''C''<sup>1</sup> isometrically embedded into an arbitrarily small [[ball (mathematics)|&epsilon;-ball]] in Euclidean 3-space (for small <math>\epsilon</math> there is no such ''C''<sup>2</sup>-embedding since from the [[Gaussian_curvature#Alternative_Formulas|formula for the Gauss curvature]] an extremal point of such an embedding would have curvature ≥ &epsilon;<sup>-2</sup>).  And, there exist ''C''<sup>1</sup> isometric embeddings of the hyperbolic plane in '''R'''<sup>3</sup>.

==''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 [[injective]] map ƒ: ''M'' → '''R'''<sup>''n''</sup> (also analytic or of class ''C<sup>k</sup>'') such that 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).

In a later [http://web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/Erratum.txt conversation with Robert M. Solovay], Nash mentioned of a fault in the original argument in deriving the sufficing value of the dimension of the embedding space for the case of non-compact manifolds.

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.{{Citation needed|date=November 2011}} 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==
* {{citation|last=Greene|first=Robert E.|last2 = Jacobowitz|first2=Howard|title= Analytic Isometric Embeddings|journal=[[Annals of Mathematics]]|volume=93|pages=189–204|doi=10.2307/1970760|issue=1|year=1971|jstor=1970760}}
* {{citation|first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash [On the embedding theorem of J. Nash]|
journal=[[Mathematische Nachrichten]]|volume= 144 |year=1989|pages= 165–187|doi=10.1002/mana.19891440113}}
*{{citation | last1 = Han|first1=Qing|last2=Hong|first2=Jia-Xing | title = Isometric Embedding of Riemannian Manifolds in Euclidean Spaces | publisher = American Mathematical Society | year = 2006 | isbn= 0-8218-4071-1}}
* {{citation|first=N.H.|last=Kuiper|authorlink=Nicolaas Kuiper|title=On ''C''<sup>1</sup>-isometric imbeddings I|journal=Nederl. Akad. Wetensch. Proc. Ser. A.|volume=58|year=1955|pages=545–556}}.
* {{citation|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|title=''C''<sup>1</sup>-isometric imbeddings|journal=[[Annals of Mathematics]]|volume=60|year=1954|pages=383–396|doi=10.2307/1969840|issue=3|jstor=1969840}}.
* {{citation|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|title=The imbedding problem for Riemannian manifolds|journal=[[Annals of Mathematics]]|volume=63|year=1956|pages=20–63|doi=10.2307/1969989|issue=1|mr=0075639|jstor=1969989}}.
* {{citation|first=John|last=Nash|title=Analyticity of the solutions of implicit function problem with analytic data|authorlink=John Forbes Nash, Jr.|journal=[[Annals of Mathematics]]|volume=84|year=1966|pages=345–355|doi=10.2307/1970448|issue=3|jstor=1970448}}.

[[Category:Theorems in Riemannian geometry]]