Mixing
What exactly is in the standart atlas of $R^n$?
$begingroup$
I saw a theorem saying that any atlas on a manifold can be extended to a unique maximal atlas.
Concerning the standard atlas of $mathbb{R}^n$, we saw that this was the atlas generated by the chart $(mathbb{R^n}, Id_{mathbb{R}^n})$. So what does it contain? My guess is that it contains every diffeomorphism (here by diffeomorphism I mean the usual notion of diffeomorphism of $mathbb{R^n}$) (but on what sets???). Indeed contains $(mathbb{R^n}, Id_{mathbb{R}^n})$ and it is smoothly compatible since a composition of diffeomorpism is a diffeomorphism.
differential-geometry smooth-manifolds
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
$endgroup$
add a comment |
$begingroup$
I saw a theorem saying that any atlas on a manifold can be extended to a unique maximal atlas.
Concerning the standard atlas of $mathbb{R}^n$, we saw that this was the atlas generated by the chart $(mathbb{R^n}, Id_{mathbb{R}^n})$. So what does it contain? My guess is that it contains every diffeomorphism (here by diffeomorphism I mean the usual notion of diffeomorphism of $mathbb{R^n}$) (but on what sets???). Indeed contains $(mathbb{R^n}, Id_{mathbb{R}^n})$ and it is smoothly compatible since a composition of diffeomorpism is a diffeomorphism.
differential-geometry smooth-manifolds
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
$endgroup$
1
$begingroup$
Your guess is correct - every "usual" diffeomorphism (on every open subset of $mathbb{R}^n$) - these are exactly all the charts compatible with the standard one.
$endgroup$
– Pig
Jan 14 at 1:49
add a comment |
$begingroup$
I saw a theorem saying that any atlas on a manifold can be extended to a unique maximal atlas.
Concerning the standard atlas of $mathbb{R}^n$, we saw that this was the atlas generated by the chart $(mathbb{R^n}, Id_{mathbb{R}^n})$. So what does it contain? My guess is that it contains every diffeomorphism (here by diffeomorphism I mean the usual notion of diffeomorphism of $mathbb{R^n}$) (but on what sets???). Indeed contains $(mathbb{R^n}, Id_{mathbb{R}^n})$ and it is smoothly compatible since a composition of diffeomorpism is a diffeomorphism.
differential-geometry smooth-manifolds
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
$endgroup$
I saw a theorem saying that any atlas on a manifold can be extended to a unique maximal atlas.
Concerning the standard atlas of $mathbb{R}^n$, we saw that this was the atlas generated by the chart $(mathbb{R^n}, Id_{mathbb{R}^n})$. So what does it contain? My guess is that it contains every diffeomorphism (here by diffeomorphism I mean the usual notion of diffeomorphism of $mathbb{R^n}$) (but on what sets???). Indeed contains $(mathbb{R^n}, Id_{mathbb{R}^n})$ and it is smoothly compatible since a composition of diffeomorpism is a diffeomorphism.
differential-geometry smooth-manifolds
differential-geometry smooth-manifolds
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
asked Jan 13 at 22:02
roi_saumonroi_saumon
56938
56938
1
$begingroup$
Your guess is correct - every "usual" diffeomorphism (on every open subset of $mathbb{R}^n$) - these are exactly all the charts compatible with the standard one.
$endgroup$
– Pig
Jan 14 at 1:49
add a comment |
1
$begingroup$
Your guess is correct - every "usual" diffeomorphism (on every open subset of $mathbb{R}^n$) - these are exactly all the charts compatible with the standard one.
$endgroup$
– Pig
Jan 14 at 1:49
1
1
$begingroup$
Your guess is correct - every "usual" diffeomorphism (on every open subset of $mathbb{R}^n$) - these are exactly all the charts compatible with the standard one.
$endgroup$
– Pig
Jan 14 at 1:49
$begingroup$
Your guess is correct - every "usual" diffeomorphism (on every open subset of $mathbb{R}^n$) - these are exactly all the charts compatible with the standard one.
$endgroup$
– Pig
Jan 14 at 1:49
add a comment |
1 Answer
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The standard atlas of $mathbb{R}^n$ just contains one chart: the set $mathbb{R}^n$ with the map $Id_{mathbb{R}^n}$, which is a diffeomorphism. Since you have only one chart you don't have to care about transition maps.
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
$endgroup$
$begingroup$
I think that in the context of the question the "standard" atlas is the unique maximal atlas that contains the identity chart, so the question (answered in the comment by @Pig) is about what is in that atlas.
$endgroup$
– Lee Mosher
Jan 14 at 23:17
add a comment |
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1 Answer
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$begingroup$
The standard atlas of $mathbb{R}^n$ just contains one chart: the set $mathbb{R}^n$ with the map $Id_{mathbb{R}^n}$, which is a diffeomorphism. Since you have only one chart you don't have to care about transition maps.
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
$endgroup$
$begingroup$
I think that in the context of the question the "standard" atlas is the unique maximal atlas that contains the identity chart, so the question (answered in the comment by @Pig) is about what is in that atlas.
$endgroup$
– Lee Mosher
Jan 14 at 23:17
add a comment |
$begingroup$
The standard atlas of $mathbb{R}^n$ just contains one chart: the set $mathbb{R}^n$ with the map $Id_{mathbb{R}^n}$, which is a diffeomorphism. Since you have only one chart you don't have to care about transition maps.
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
$endgroup$
$begingroup$
I think that in the context of the question the "standard" atlas is the unique maximal atlas that contains the identity chart, so the question (answered in the comment by @Pig) is about what is in that atlas.
$endgroup$
– Lee Mosher
Jan 14 at 23:17
add a comment |
$begingroup$
The standard atlas of $mathbb{R}^n$ just contains one chart: the set $mathbb{R}^n$ with the map $Id_{mathbb{R}^n}$, which is a diffeomorphism. Since you have only one chart you don't have to care about transition maps.
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
$endgroup$
The standard atlas of $mathbb{R}^n$ just contains one chart: the set $mathbb{R}^n$ with the map $Id_{mathbb{R}^n}$, which is a diffeomorphism. Since you have only one chart you don't have to care about transition maps.
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
answered Jan 13 at 22:25
SchlubbidubbiSchlubbidubbi
1585
1585
$begingroup$
I think that in the context of the question the "standard" atlas is the unique maximal atlas that contains the identity chart, so the question (answered in the comment by @Pig) is about what is in that atlas.
$endgroup$
– Lee Mosher
Jan 14 at 23:17
add a comment |
$begingroup$
I think that in the context of the question the "standard" atlas is the unique maximal atlas that contains the identity chart, so the question (answered in the comment by @Pig) is about what is in that atlas.
$endgroup$
– Lee Mosher
Jan 14 at 23:17
$begingroup$
I think that in the context of the question the "standard" atlas is the unique maximal atlas that contains the identity chart, so the question (answered in the comment by @Pig) is about what is in that atlas.
$endgroup$
– Lee Mosher
Jan 14 at 23:17
$begingroup$
I think that in the context of the question the "standard" atlas is the unique maximal atlas that contains the identity chart, so the question (answered in the comment by @Pig) is about what is in that atlas.
$endgroup$
– Lee Mosher
Jan 14 at 23:17
add a comment |
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$begingroup$
Your guess is correct - every "usual" diffeomorphism (on every open subset of $mathbb{R}^n$) - these are exactly all the charts compatible with the standard one.
$endgroup$
– Pig
Jan 14 at 1:49