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'embedding'
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| Title of object: |
embedding |
| Canonical Name: |
Embedding3 |
| Type: |
Definition |
| Created on: |
2004-12-11 01:00:42 |
| Modified on: |
2004-12-11 01:07:04 |
| Classification: |
msc:57R40 |
| Defines: |
Whitney's theorem |
Preamble:
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\usepackage{amssymb,amscd}
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%\usepackage{psfrag}
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Content:
\PMlinkescapeword{characterization}
\PMlinkescapeword{states}
Let $M$ and $N$ be manifolds and $f\colon M\rightarrow N$ a smooth map. Then $f$ is an \emph{embedding} if
\begin{enumerate}
\item $f(M)$ is a submanifold of $N$, and
\item by abuse of notation, $f\colon M\rightarrow f(M)$ is a diffeomorphism.
\end{enumerate}
The above characterization can be equivalently stated:
$f\colon M\rightarrow N$ is an embedding if
\begin{enumerate}
\item $f$ is an immersion, and
\item by abuse of notation, $f\colon M\rightarrow f(M)$ is a homeomorphism.
\end{enumerate}
\textbf{Remark}. A celebrated theorem of Whitney states that every $n$ dimensional manifold admits an embedding into $\mathbb{R}^{2n+1}$. |
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