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.
Facts
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. Connections
Sources
1. Nash Embedding Theorem (Wikipedia)
Wikimedia FoundationLead sectionQuote, Lead section
The C1 theorem was published in 1954, and the Ck theorem in 1956.
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