Nash embedding theorems

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. The first theorem is for continuously differentiable (C1) embeddings and the second for embeddings that are analytic or smooth of class Ck, 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 C1 theorem was published in 1954, the Ck-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by . (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 Ck- case was later extrapolated into the h-principle and Nash–Moser implicit function theorem. A simpler proof of the second Nash embedding theorem was obtained by who reduced the set of nonlinear partial differential equations to an elliptic system, to which the contraction mapping theorem could be applied. Given an m-dimensional Riemannian manifold (M, g), an isometric embedding is a continuously differentiable topological embedding f: M → Rn such that the pullback of the Euclidean metric equals g. In analytical terms, this may be viewed (relative to a smooth coordinate chart x) as a system of 1/2m(m + 1) many first-order partial differential equations for n unknown (real-valued) functions: If n is less than 1/2m(m + 1), then there are more equations than unknowns.