Smaller neighborhood around the identity of Lie Group The Next CEO of Stack OverflowAbout connected Lie GroupsIs $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?Map $n(g,h) = gh^-1$ is smooth implies $G$ is a Lie Group.(Whitney) Extension Lemma for smooth mapsImmersed subgroup of a Lie group is a Lie group?Fundamental theorem on flows lee's book 2nd edition$G$ a Lie group, $V, S$ submanifolds of $G$ containing $e$, $psi : V times S rightarrow G$; $psi(v,s)=vs$, then $dpsi(X,0)=X$ and $dpsi(0,Y)=Y$Open neighborhood in Lie groupfactoring a neighborhood of identity in a compact connected Lie group with a closed Lie subgroupJohn Lee : Cubical charts and cube in $mathbbR^n$
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Smaller neighborhood around the identity of Lie Group
The Next CEO of Stack OverflowAbout connected Lie GroupsIs $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?Map $n(g,h) = gh^-1$ is smooth implies $G$ is a Lie Group.(Whitney) Extension Lemma for smooth mapsImmersed subgroup of a Lie group is a Lie group?Fundamental theorem on flows lee's book 2nd edition$G$ a Lie group, $V, S$ submanifolds of $G$ containing $e$, $psi : V times S rightarrow G$; $psi(v,s)=vs$, then $dpsi(X,0)=X$ and $dpsi(0,Y)=Y$Open neighborhood in Lie groupfactoring a neighborhood of identity in a compact connected Lie group with a closed Lie subgroupJohn Lee : Cubical charts and cube in $mathbbR^n$
$begingroup$
This is the problem 7-6 of Lee's Introduction to Smooth Manifolds (2nd edition):
Suppose G is a Lie group and U is any neighborhood of the identity. Show
that there exists a neighborhood V of the identity such that $V subset U$ and $gh^-1 in U$ whenever $g, h in V$.
How do I approach this problem? I tried using the smoothness of $(g,h) mapsto gh^-1$ or the open subgroup generated by $U$, but it didn't get me anywhere.
differential-geometry manifolds lie-groups smooth-manifolds
edited Mar 19 at 7:51
Sou
3,3242923
asked Mar 19 at 4:40
o zo z
132
$endgroup$
add a comment |
$begingroup$
This is the problem 7-6 of Lee's Introduction to Smooth Manifolds (2nd edition):
Suppose G is a Lie group and U is any neighborhood of the identity. Show
that there exists a neighborhood V of the identity such that $V subset U$ and $gh^-1 in U$ whenever $g, h in V$.
How do I approach this problem? I tried using the smoothness of $(g,h) mapsto gh^-1$ or the open subgroup generated by $U$, but it didn't get me anywhere.
differential-geometry manifolds lie-groups smooth-manifolds
edited Mar 19 at 7:51
Sou
3,3242923
asked Mar 19 at 4:40
o zo z
132
$endgroup$
add a comment |
$begingroup$
This is the problem 7-6 of Lee's Introduction to Smooth Manifolds (2nd edition):
Suppose G is a Lie group and U is any neighborhood of the identity. Show
that there exists a neighborhood V of the identity such that $V subset U$ and $gh^-1 in U$ whenever $g, h in V$.
How do I approach this problem? I tried using the smoothness of $(g,h) mapsto gh^-1$ or the open subgroup generated by $U$, but it didn't get me anywhere.
differential-geometry manifolds lie-groups smooth-manifolds
edited Mar 19 at 7:51
Sou
3,3242923
asked Mar 19 at 4:40
o zo z
132
$endgroup$
This is the problem 7-6 of Lee's Introduction to Smooth Manifolds (2nd edition):
Suppose G is a Lie group and U is any neighborhood of the identity. Show
that there exists a neighborhood V of the identity such that $V subset U$ and $gh^-1 in U$ whenever $g, h in V$.
How do I approach this problem? I tried using the smoothness of $(g,h) mapsto gh^-1$ or the open subgroup generated by $U$, but it didn't get me anywhere.
differential-geometry manifolds lie-groups smooth-manifolds
differential-geometry manifolds lie-groups smooth-manifolds
edited Mar 19 at 7:51
Sou
3,3242923
asked Mar 19 at 4:40
o zo z
132
edited Mar 19 at 7:51
Sou
3,3242923
asked Mar 19 at 4:40
o zo z
132
edited Mar 19 at 7:51
Sou
3,3242923
edited Mar 19 at 7:51
Sou
3,3242923
edited Mar 19 at 7:51
Sou
3,3242923
3,3242923
asked Mar 19 at 4:40
o zo z
132
asked Mar 19 at 4:40
o zo z
132
asked Mar 19 at 4:40
o zo z
132
132
add a comment |
add a comment |
1 Answer
1
active
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votes
$begingroup$
You don’t need anything fancy here. Just use continuity and pick $V$ carefully.
The map $f : G times G to G$ defined as $f(g,h) = gh^-1$ is a smooth map. Suppose $U$ is a neighbourhood of the identity $ein G$. By continuity, $W = f^-1(U)$ is open in $G times G$. Since $(e,e) in W$, there are neighbourhoods $W_1,W_2 subseteq G$ containing $ein G$ such that $W_1times W_2 subseteq W$. Choose $V$ as
$$V = (W_1 cap W_2) cap U$$
So $V times V subseteq W=f^-1(U)$ implies $f(V times V) subseteq U$. I.e., $forall g,h in V$, we have $f(g,h)=gh^-1 in U$.
answered Mar 19 at 7:47
SouSou
3,3242923
$endgroup$
add a comment |
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$begingroup$
You don’t need anything fancy here. Just use continuity and pick $V$ carefully.
The map $f : G times G to G$ defined as $f(g,h) = gh^-1$ is a smooth map. Suppose $U$ is a neighbourhood of the identity $ein G$. By continuity, $W = f^-1(U)$ is open in $G times G$. Since $(e,e) in W$, there are neighbourhoods $W_1,W_2 subseteq G$ containing $ein G$ such that $W_1times W_2 subseteq W$. Choose $V$ as
$$V = (W_1 cap W_2) cap U$$
So $V times V subseteq W=f^-1(U)$ implies $f(V times V) subseteq U$. I.e., $forall g,h in V$, we have $f(g,h)=gh^-1 in U$.
answered Mar 19 at 7:47
SouSou
3,3242923
$endgroup$
add a comment |
$begingroup$
You don’t need anything fancy here. Just use continuity and pick $V$ carefully.
The map $f : G times G to G$ defined as $f(g,h) = gh^-1$ is a smooth map. Suppose $U$ is a neighbourhood of the identity $ein G$. By continuity, $W = f^-1(U)$ is open in $G times G$. Since $(e,e) in W$, there are neighbourhoods $W_1,W_2 subseteq G$ containing $ein G$ such that $W_1times W_2 subseteq W$. Choose $V$ as
$$V = (W_1 cap W_2) cap U$$
So $V times V subseteq W=f^-1(U)$ implies $f(V times V) subseteq U$. I.e., $forall g,h in V$, we have $f(g,h)=gh^-1 in U$.
answered Mar 19 at 7:47
SouSou
3,3242923
$endgroup$
add a comment |
$begingroup$
You don’t need anything fancy here. Just use continuity and pick $V$ carefully.
The map $f : G times G to G$ defined as $f(g,h) = gh^-1$ is a smooth map. Suppose $U$ is a neighbourhood of the identity $ein G$. By continuity, $W = f^-1(U)$ is open in $G times G$. Since $(e,e) in W$, there are neighbourhoods $W_1,W_2 subseteq G$ containing $ein G$ such that $W_1times W_2 subseteq W$. Choose $V$ as
$$V = (W_1 cap W_2) cap U$$
So $V times V subseteq W=f^-1(U)$ implies $f(V times V) subseteq U$. I.e., $forall g,h in V$, we have $f(g,h)=gh^-1 in U$.
answered Mar 19 at 7:47
SouSou
3,3242923
$endgroup$
You don’t need anything fancy here. Just use continuity and pick $V$ carefully.
The map $f : G times G to G$ defined as $f(g,h) = gh^-1$ is a smooth map. Suppose $U$ is a neighbourhood of the identity $ein G$. By continuity, $W = f^-1(U)$ is open in $G times G$. Since $(e,e) in W$, there are neighbourhoods $W_1,W_2 subseteq G$ containing $ein G$ such that $W_1times W_2 subseteq W$. Choose $V$ as
$$V = (W_1 cap W_2) cap U$$
So $V times V subseteq W=f^-1(U)$ implies $f(V times V) subseteq U$. I.e., $forall g,h in V$, we have $f(g,h)=gh^-1 in U$.
answered Mar 19 at 7:47
SouSou
3,3242923
edited Mar 25 at 12:18
answered Mar 19 at 7:47
SouSou
3,3242923
answered Mar 19 at 7:47
SouSou
3,3242923
answered Mar 19 at 7:47
SouSou
3,3242923
3,3242923
add a comment |
add a comment |
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