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embedding (Definition)

Let $M$ and $N$ be manifolds and $f\colon M\to N$ a smooth map. Then $f$ is an embedding if

  1. $f(M)$ is a submanifold of $N$ , and
  2. $f\colon M\to f(M)$ 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

  1. $f$ is an immersion, and
  2. 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 $\mathbb{R}^{2n+1}$ .




"embedding" is owned by CWoo.
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Other names:  differential embedding
Also defines:  Whitney's theorem

Attachments:
abstract embedding (Definition) by juanman
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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 MSC57R40 (Manifolds and cell complexes :: Differential topology :: Embeddings)

Pending Errata and Addenda
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change of name by juanman on 2007-04-16 14:48:50
this article should be called differential embedding in view that there is also algebraic embedding and others
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