Embedding (simple definition)
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A ''smooth embedding'' of a smooth compact manifold $N$ into a smooth manifold is a smooth injective map $f:N\to M$ such that | A ''smooth embedding'' of a smooth compact manifold $N$ into a smooth manifold is a smooth injective map $f:N\to M$ such that | ||
$df$ is a monomorphism at each point. | $df$ is a monomorphism at each point. | ||
| − | (See an [[Embedding#Definition|alternative definition]] which works for non-compact manifolds and involves immersions. | + | (See an [[Embedding#Definition|equivalent alternative definition]] which works for non-compact manifolds and involves immersions. |
A ''smooth immersion'' is a smooth map $f:N\to M$ such that $df$ is a monomorphism at each point. | A ''smooth immersion'' is a smooth map $f:N\to M$ such that $df$ is a monomorphism at each point. | ||
| − | See an [[Immersion#Definition|alternative definition]].) | + | See an [[Immersion#Definition|equivalent alternative definition]].) |
A map $f:N\to\Rr^m$ of a polyhedron $N$ is ''piecewise-linear (PL)'' if it is linear on each simplex of some smooth triangulation of $N$. | A map $f:N\to\Rr^m$ of a polyhedron $N$ is ''piecewise-linear (PL)'' if it is linear on each simplex of some smooth triangulation of $N$. | ||
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A ''topological embedding'' of a compact subset $N\subset\Rr^n$ into $\Rr^m$ is a continuous injective map $f:N\to\Rr^m$. | A ''topological embedding'' of a compact subset $N\subset\Rr^n$ into $\Rr^m$ is a continuous injective map $f:N\to\Rr^m$. | ||
</wikitex> | </wikitex> | ||
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[[Category:Definitions]] | [[Category:Definitions]] | ||
Revision as of 09:45, 14 October 2016
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This page has not been refereed. The information given here might be incomplete or provisional. |
Definition
A smooth embedding of a smooth compact manifold 
into a smooth manifold is a smooth injective map 
such that
is a monomorphism at each point.
(See an equivalent alternative definition which works for non-compact manifolds and involves immersions.
A smooth immersion is a smooth map such that is a monomorphism at each point.
See an equivalent alternative definition.)
A map 
of a polyhedron is piecewise-linear (PL) if it is linear on each simplex of some smooth triangulation of .
A PL embedding of a compact polyhedron into 
is a PL injective map .
A topological embedding of a compact subset 
into is a continuous injective map .
