Is every local diffeomorphism from $U subseteq mathbb{R}^k$ to a $V subseteq M$ equivalent to a chart?Smooth...
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Is every local diffeomorphism from $U subseteq mathbb{R}^k$ to a $V subseteq M$ equivalent to a chart?
Smooth chart in what sense?What does “smoothly compatible” mean?Non-smooth internal function of composition which happens to be smoothIs every local diffeomorphic chart a local isometry?Smooth structure of embedded submanifoldssmooth submersions and maps with local sectionsBijective local diffeomorphism is a diffeomorphismSmooth map between vector bundles from local coordinatesEquivalence of the two definitions of smooth mapsChoosing Coordinate Charts — Smooth Functions on Manifolds
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If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?
It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.
differential-geometry smooth-manifolds
asked 2 days ago
JeffJeff
13318
$endgroup$
add a comment |
$begingroup$
If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?
It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.
differential-geometry smooth-manifolds
asked 2 days ago
JeffJeff
13318
$endgroup$
add a comment |
$begingroup$
If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?
It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.
differential-geometry smooth-manifolds
asked 2 days ago
JeffJeff
13318
$endgroup$
If we have a local diffeomorphism $F: U rightarrow V$, with $U subseteq mathbb{R}^k$ and $V subseteq M$, and $mathbb{R}^k$ with the usual smooth structure, then is it always the case that we can consider $F^{-1}$ as a chart map for $M$?
It seems so because as a diffeomorphism, $F circ psi^{-1}$ implies compatibility with the charts of $M$.
differential-geometry smooth-manifolds
differential-geometry smooth-manifolds
asked 2 days ago
JeffJeff
13318
asked 2 days ago
JeffJeff
13318
edited 2 days ago
Jeff
asked 2 days ago
JeffJeff
13318
asked 2 days ago
JeffJeff
13318
asked 2 days ago
JeffJeff
13318
13318
add a comment |
add a comment |
1 Answer
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Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).
Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
phi circ Fvert_{W cap X}^{-1})$ is a chart for $M$.
In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.
answered 2 days ago
setnosetsetnoset
36518
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1 Answer
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1 Answer
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$begingroup$
Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).
Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
phi circ Fvert_{W cap X}^{-1})$ is a chart for $M$.
In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.
answered 2 days ago
setnosetsetnoset
36518
$endgroup$
add a comment |
$begingroup$
Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).
Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
phi circ Fvert_{W cap X}^{-1})$ is a chart for $M$.
In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.
answered 2 days ago
setnosetsetnoset
36518
$endgroup$
add a comment |
$begingroup$
Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).
Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
phi circ Fvert_{W cap X}^{-1})$ is a chart for $M$.
In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.
answered 2 days ago
setnosetsetnoset
36518
$endgroup$
Given any point $p$ in $F(U) subseteq V$ and $x in F^{-1}({p})$ there exists a neighbourhood $W subseteq U $ around $x$ such that $F(W)$ is open with respect to $U$ and $F vert_W: W rightarrow F(W)$ is a diffeomorphism (by definition of local diffeomorphism).
Now we have a problem, since there is no guarantee that $W$ is open with respect to $mathbb{R}^k$. Regardless, if $(X,phi)$ is a chart for $U$ (seeing $U$ as an embedded submanifold of $mathbb{R}^k$) with $x in X$, then $(F(W cap X),
phi circ Fvert_{W cap X}^{-1})$ is a chart for $M$.
In the special case where $U$ is open, $W$ is also open in $mathbb{R}^k$, so that $(F(W), Fvert_W^{-1})$ is a chart for $M$.
answered 2 days ago
setnosetsetnoset
36518
edited 2 days ago
answered 2 days ago
setnosetsetnoset
36518
answered 2 days ago
setnosetsetnoset
36518
answered 2 days ago
setnosetsetnoset
36518
36518
add a comment |
add a comment |
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