Loterre: Mathematics: Nash embedding theorem

geometry > differential geometry > Riemannian geometry > Nash embedding theorem

Nash embedding theorem

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every 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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Nash_embedding_theorems)

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