Nash isometric embedding theorem

Every compact $n$ -dimensional Riemannian manifold $M$ of class $C^{k}$ ( $3\\leq k\\leq\\infty$ ) can be $C^{k}$ -isometrically imbedded in any smallportion of a Euclidean space $\\mathbb{R}^{N}$ , where

N=\\frac{1}{2}n(3n+11).

Every non-compact $n$ -dimensional Riemannian manifold $M$ of class $C^{k}$ ( $3\\leq k\\leq\\infty$ ) can be $C^{k}$ -isometrically imbedded in any small portion of a Euclidean space $\\mathbb{R}^{N}$ , where

N=(n+1)\\frac{1}{2}n(3n+11).

The original proof due to Nash relying on an iteration scheme has been considerably simplified. For an overview, see [2].

References

1 Nash, J. F., The imbedding problem for Riemannian manifold, Ann. of Math. 63 (1956), 20–63 (MR 17, 782)
2 D. Yang, Gunther’s proof of Nash’s isometric embedding theorem, http://www.math.poly.edu/ yang/papers/gunther.pdfonline