PlanetMath: embedding

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embedding

(Definition)

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

is a submanifold of , 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 is a diffeomorphism.)

The above characterization can be equivalently stated: $f\colon M\to N$ is an embedding if

Remark. A celebrated theorem of Whitney states that every 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

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Cross-references: homeomorphism, immersion, map, diffeomorphism, submanifold, smooth map, manifolds
There are 8 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 4841 times total.

Classification:

57R40 (Manifolds and cell complexes :: Differential topology :: Embeddings)

Pending Errata and Addenda

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