Editing Nash embedding theorems (section)

{{Short description|Every Riemannian manifold can be isometrically embedded into some Euclidean space}}

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]] or [[smooth function|smooth]] of class ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result.

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.{{sfnm|1a1=Taylor|1y=2011|1pp=147–151}}

{{Anchor|Nash–Kuiper theorem}}