embedding

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

Remark. A celebrated theorem of Whitney states that every $n$ dimensional manifold admits an embedding into $\mathbb{R}^{2n+1}$ .