# embedding

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

1. 1.

$f(M)$ is a submanifold of $N$, and

2. 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. 1.

$f$ is an immersion, and

2. 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 $\mathbb{R}^{2n+1}$.

Title embedding Embedding 2013-03-22 14:52:46 2013-03-22 14:52:46 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 57R40 differential embedding Whitney’s theorem