embedding

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

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

2. $f:M\to f(M)$ is a diffeomorphism. (There’s an abuse of notation here. This should really be restated as the map $g:M\to f(M)$ defined by $g(p)=f(p)$ is a diffeomorphism.)

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

1. $f$ is an immersion, and

2. by abuse of notation, $f:M\to f(M)$ is a homeomorphism.

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

Title	embedding
Canonical name	Embedding
Date of creation	2013-03-22 14:52:46
Last modified on	2013-03-22 14:52:46
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Definition
Classification	msc 57R40
Synonym	differential embedding
Defines	Whitney’s theorem