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Nash isometric embedding theorem
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(Theorem)
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Every compact  -dimensional Riemannian manifold  of class  (
 ) can be  -isometrically imbedded in any small portion of a Euclidean space
 , where
Every non-compact  -dimensional Riemannian manifold  of class  (
 ) can be  -isometrically imbedded in any small portion of a Euclidean space
 , where
The original proof due to Nash relying on an iteration scheme has been considerably simplified. For an overview, see [2].
- 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, online
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"Nash isometric embedding theorem" is owned by Simone. [ full author list (4) ]
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Cross-references: scheme, iteration, proof, Euclidean space, class, Riemannian manifold, compact
This is version 4 of Nash isometric embedding theorem, born on 2006-01-21, modified 2007-10-06.
Object id is 7569, canonical name is NashIsometricEmbeddingTheorem.
Accessed 2209 times total.
Classification:
| AMS MSC: | 53C20 (Differential geometry :: Global differential geometry :: Global Riemannian geometry, including pinching) | | | 53C42 (Differential geometry :: Global differential geometry :: Immersions ) | | | 57R40 (Manifolds and cell complexes :: Differential topology :: Embeddings) | | | 58A05 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differentiable manifolds, foundations) |
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Pending Errata and Addenda
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