# 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 small portion 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
Title Nash isometric embedding theorem NashIsometricEmbeddingTheorem 2013-03-22 15:38:17 2013-03-22 15:38:17 Simone (5904) Simone (5904) 7 Simone (5904) Theorem msc 53C20 msc 53C42 msc 57R40 msc 58A05