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m Facts707 moved page Nash embedding theorem to Nash embedding theorems over redirect: plural |
John Forbes Nash Jr. |
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{{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 [[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 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.
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