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Let $M$ and $N$ be manifolds and $f\colon M\to N$ a smooth map. Then $f$ is an embedding if
- $f(M)$ is a submanifold of $N$ , and
- $f\colon M\to f(M)$ is a diffeomorphism. (There's an abuse of notation here. This should really be restated as the map $g\colon M\to f(M)$ defined by $g(p)=f(p)$ is a diffeomorphism.)
The above characterization can be equivalently stated: $f\colon M\to N$ is an embedding if
- $f$ is an immersion, and
- by abuse of notation, $f\colon M\to f(M)$ is a homeomorphism.
Remark. A celebrated theorem of Whitney states that every $n$ dimensional manifold admits an embedding into $\mathbb{R}^{2n+1}$ .
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"embedding" is owned by CWoo.
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(view preamble | get metadata)
| Other names: |
differential embedding |
| Also defines: |
Whitney's theorem |
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Cross-references: theorem, homeomorphism, immersion, map, diffeomorphism, submanifold, smooth map, manifolds
There are 16 references to this entry.
This is version 5 of embedding, born on 2004-12-11, modified 2007-04-16.
Object id is 6557, canonical name is Embedding3.
Accessed 6380 times total.
Classification:
| AMS MSC: | 57R40 (Manifolds and cell complexes :: Differential topology :: Embeddings) |
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Pending Errata and Addenda
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