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'Nash isometric embedding theorem'
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| Title of object: |
Nash isometric embedding theorem |
| Canonical Name: |
NashIsometricEmbeddingTheorem |
| Type: |
Theorem |
| Created on: |
2006-01-21 13:11:58 |
| Modified on: |
2006-01-21 13:45:29 |
| Classification: |
msc:53C42 |
Preamble:
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Content:
Every compact $n$-dimensional Riemannian manifold $M$ of class $C^k$ ($3\le k\le\infty$) can be $C^k$-isometrically imbedded in any small portion of a Euclidean space $\mathbb R^N$, where $N=\frac 12 n(3n+11)$.\, Every non-compact $n$-dimensional Riemannian manifold $M$ of class $C^k$ ($3\le k\le\infty$) can be $C^k$-isometrically imbedded in any small portion of a Euclidean space $\mathbb R^N$, where $N=\frac 12 n(n+1)(3n+11)$.
\begin{thebibliography}
{}Nash, J. F., \emph{The imbedding problem for Riemannian manifold}, Ann. of Math. 63 (1956), 20--63 (MR 17, 782)
\end{thebibliography} |
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