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

Defines:

Whitney's theorem

Synonym:

differential embedding

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

57R40*no label found*

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## Comments

## change of name

this article should be called differential embedding in view that there is also algebraic embedding and others

## Re: change of name

I have added "differential embedding" as a synonym, rather than changin the title, because "embedding" is a common term in the literature.