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The '''Nash embedding theorems''' (or '''imbedding theorems'''), named after [[John Forbes Nash Jr.]], state that every [[Riemannian manifold]] can be isometrically [[embedding|embedded]] into some [[Euclidean space]]. [[Isometry|Isometric]] means preserving the length of every [[rectifiable path|path]]. For instance, bending but neither stretching nor 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 embeddings that are [[analytic function|analytic]]
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 simpler 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.<ref>{{cite book|first=Michael E.|last=Taylor|author-link=Michael E. Taylor|title=Partial Differential Equations III: Nonlinear equations|mr=2744149 |edition = 2nd |series=Applied Mathematical Sciences|volume= 117|publisher= Springer|year= 2011|isbn=978-1-4419-7048-0|chapter=Isometric imbedding of Riemannian manifolds|pages=147–151}}</ref>
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