Difference between a parametrized surface and manifold
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What is the difference between a parametrized surface and manifold?
Is it true that if $M subset mathbb R^n$ is an $n$-dimensional parametrized surface it is also a (parametrized?) manifold? I am stuck with understanding the concept of a manifold. In which cases would a parametrized surface not be parametrized manifold or vice versa?
manifolds parametrization
asked Jan 20 at 9:42
TeslaTesla
885426
$endgroup$
add a comment |
$begingroup$
What is the difference between a parametrized surface and manifold?
Is it true that if $M subset mathbb R^n$ is an $n$-dimensional parametrized surface it is also a (parametrized?) manifold? I am stuck with understanding the concept of a manifold. In which cases would a parametrized surface not be parametrized manifold or vice versa?
manifolds parametrization
asked Jan 20 at 9:42
TeslaTesla
885426
$endgroup$
1
$begingroup$
Roughly speaking, manifolds are kind of like the union of several parametrized surfaces with certain restrictions when the surfaces overlap.
$endgroup$
– xbh
Jan 20 at 9:49
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So if I have a family of $(S_i)_i$ of parametrized surfaces, then the manifold is $bigcup_i S_i $, but overlaps "count only once"?
$endgroup$
– Tesla
Jan 20 at 13:34
$begingroup$
As a set, yes. But generally we require the parametrizations be consistent on the overlapped part, so maybe not all family could become manifolds by union.
$endgroup$
– xbh
Jan 20 at 13:39
add a comment |
$begingroup$
What is the difference between a parametrized surface and manifold?
Is it true that if $M subset mathbb R^n$ is an $n$-dimensional parametrized surface it is also a (parametrized?) manifold? I am stuck with understanding the concept of a manifold. In which cases would a parametrized surface not be parametrized manifold or vice versa?
manifolds parametrization
asked Jan 20 at 9:42
TeslaTesla
885426
$endgroup$
What is the difference between a parametrized surface and manifold?
Is it true that if $M subset mathbb R^n$ is an $n$-dimensional parametrized surface it is also a (parametrized?) manifold? I am stuck with understanding the concept of a manifold. In which cases would a parametrized surface not be parametrized manifold or vice versa?
manifolds parametrization
manifolds parametrization
asked Jan 20 at 9:42
TeslaTesla
885426
asked Jan 20 at 9:42
TeslaTesla
885426
asked Jan 20 at 9:42
TeslaTesla
885426
asked Jan 20 at 9:42
TeslaTesla
885426
asked Jan 20 at 9:42
TeslaTesla
885426
885426
1
$begingroup$
Roughly speaking, manifolds are kind of like the union of several parametrized surfaces with certain restrictions when the surfaces overlap.
$endgroup$
– xbh
Jan 20 at 9:49
$begingroup$
So if I have a family of $(S_i)_i$ of parametrized surfaces, then the manifold is $bigcup_i S_i $, but overlaps "count only once"?
$endgroup$
– Tesla
Jan 20 at 13:34
$begingroup$
As a set, yes. But generally we require the parametrizations be consistent on the overlapped part, so maybe not all family could become manifolds by union.
$endgroup$
– xbh
Jan 20 at 13:39
add a comment |
1
$begingroup$
Roughly speaking, manifolds are kind of like the union of several parametrized surfaces with certain restrictions when the surfaces overlap.
$endgroup$
– xbh
Jan 20 at 9:49
$begingroup$
So if I have a family of $(S_i)_i$ of parametrized surfaces, then the manifold is $bigcup_i S_i $, but overlaps "count only once"?
$endgroup$
– Tesla
Jan 20 at 13:34
$begingroup$
As a set, yes. But generally we require the parametrizations be consistent on the overlapped part, so maybe not all family could become manifolds by union.
$endgroup$
– xbh
Jan 20 at 13:39
1
1
$begingroup$
Roughly speaking, manifolds are kind of like the union of several parametrized surfaces with certain restrictions when the surfaces overlap.
$endgroup$
– xbh
Jan 20 at 9:49
$begingroup$
Roughly speaking, manifolds are kind of like the union of several parametrized surfaces with certain restrictions when the surfaces overlap.
$endgroup$
– xbh
Jan 20 at 9:49
$begingroup$
So if I have a family of $(S_i)_i$ of parametrized surfaces, then the manifold is $bigcup_i S_i $, but overlaps "count only once"?
$endgroup$
– Tesla
Jan 20 at 13:34
$begingroup$
So if I have a family of $(S_i)_i$ of parametrized surfaces, then the manifold is $bigcup_i S_i $, but overlaps "count only once"?
$endgroup$
– Tesla
Jan 20 at 13:34
$begingroup$
As a set, yes. But generally we require the parametrizations be consistent on the overlapped part, so maybe not all family could become manifolds by union.
$endgroup$
– xbh
Jan 20 at 13:39
$begingroup$
As a set, yes. But generally we require the parametrizations be consistent on the overlapped part, so maybe not all family could become manifolds by union.
$endgroup$
– xbh
Jan 20 at 13:39
add a comment |
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$begingroup$
Roughly speaking, manifolds are kind of like the union of several parametrized surfaces with certain restrictions when the surfaces overlap.
$endgroup$
– xbh
Jan 20 at 9:49
$begingroup$
So if I have a family of $(S_i)_i$ of parametrized surfaces, then the manifold is $bigcup_i S_i $, but overlaps "count only once"?
$endgroup$
– Tesla
Jan 20 at 13:34
$begingroup$
As a set, yes. But generally we require the parametrizations be consistent on the overlapped part, so maybe not all family could become manifolds by union.
$endgroup$
– xbh
Jan 20 at 13:39