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

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