Nash isometric embedding theorem
Every compact^{} $n$-dimensional Riemannian manifold^{} $M$ of class ${C}^{k}$ ($3\le k\le \mathrm{\infty}$) 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 ${C}^{k}$ ($3\le k\le \mathrm{\infty}$) 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 |
---|---|
Canonical name | NashIsometricEmbeddingTheorem |
Date of creation | 2013-03-22 15:38:17 |
Last modified on | 2013-03-22 15:38:17 |
Owner | Simone (5904) |
Last modified by | Simone (5904) |
Numerical id | 7 |
Author | Simone (5904) |
Entry type | Theorem |
Classification | msc 53C20 |
Classification | msc 53C42 |
Classification | msc 57R40 |
Classification | msc 58A05 |