Every Riemannian manifold can be isometrically embedded into some Euclidean space of sufficiently high dimension. Proved by John Nash, it showed that abstract Riemannian geometry, defined intrinsically, always has a concrete realization as an ordinary curved surface or hypersurface.
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StatementEvery Riemannian manifold can be isometrically embedded into some Euclidean space, preserving the length of every path drawn on the manifold. 1 Proof YearThe continuously-differentiable (C1) version was published in 1954; the smooth (Ck) version, the one usually meant by the Nash embedding theorem, was published in 1956. Classification
Statement Form Nash Embedding Theorem
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Source Wikipedia: John Forbes Nash Jr.
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1. Nash Embedding Theorem (Wikipedia)
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The C1 theorem was published in 1954, and the Ck theorem in 1956.
Wikipedia: John Forbes Nash Jr.
WikipediaPrincipal Result Of: John Nash, Article textQuote, Principal Result Of: John Nash, Article text
In the 1950s, Nash discovered and proved the Nash embedding theorems by solving a system of nonlinear partial differential equations arising in Riemannian geometry.
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