Given $E:Nto M$ an embedding and $V,Win mathfrak{X}(M)$ tangent to $N$, we claim that the commutator of $V$...
$begingroup$
I have encounter some difficulties while looking at an exercise online. It basically goes as follows:
Given $E:Nto M$ an embedding and $V,Win mathfrak{X}(M)$ tangent to $N$, we claim that the commutator of $V$ and $W$ is also tangent to $N$.
I would like to have some ideas about how to attack the problem effectively.
Thank you in advance!
differential-geometry smooth-manifolds vector-fields
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
$endgroup$
add a comment |
$begingroup$
I have encounter some difficulties while looking at an exercise online. It basically goes as follows:
Given $E:Nto M$ an embedding and $V,Win mathfrak{X}(M)$ tangent to $N$, we claim that the commutator of $V$ and $W$ is also tangent to $N$.
I would like to have some ideas about how to attack the problem effectively.
Thank you in advance!
differential-geometry smooth-manifolds vector-fields
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
$endgroup$
$begingroup$
There are a few possible approaches, depending on your definition of the commutator.
$endgroup$
– Amitai Yuval
Jan 23 at 6:49
$begingroup$
It is just the usual one: $[A,B]=AB-BA$.
$endgroup$
– Le Théoricien.
Jan 23 at 11:35
add a comment |
$begingroup$
I have encounter some difficulties while looking at an exercise online. It basically goes as follows:
Given $E:Nto M$ an embedding and $V,Win mathfrak{X}(M)$ tangent to $N$, we claim that the commutator of $V$ and $W$ is also tangent to $N$.
I would like to have some ideas about how to attack the problem effectively.
Thank you in advance!
differential-geometry smooth-manifolds vector-fields
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
$endgroup$
I have encounter some difficulties while looking at an exercise online. It basically goes as follows:
Given $E:Nto M$ an embedding and $V,Win mathfrak{X}(M)$ tangent to $N$, we claim that the commutator of $V$ and $W$ is also tangent to $N$.
I would like to have some ideas about how to attack the problem effectively.
Thank you in advance!
differential-geometry smooth-manifolds vector-fields
differential-geometry smooth-manifolds vector-fields
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
asked Jan 23 at 5:34
Le Théoricien.Le Théoricien.
789
789
$begingroup$
There are a few possible approaches, depending on your definition of the commutator.
$endgroup$
– Amitai Yuval
Jan 23 at 6:49
$begingroup$
It is just the usual one: $[A,B]=AB-BA$.
$endgroup$
– Le Théoricien.
Jan 23 at 11:35
add a comment |
$begingroup$
There are a few possible approaches, depending on your definition of the commutator.
$endgroup$
– Amitai Yuval
Jan 23 at 6:49
$begingroup$
It is just the usual one: $[A,B]=AB-BA$.
$endgroup$
– Le Théoricien.
Jan 23 at 11:35
$begingroup$
There are a few possible approaches, depending on your definition of the commutator.
$endgroup$
– Amitai Yuval
Jan 23 at 6:49
$begingroup$
There are a few possible approaches, depending on your definition of the commutator.
$endgroup$
– Amitai Yuval
Jan 23 at 6:49
$begingroup$
It is just the usual one: $[A,B]=AB-BA$.
$endgroup$
– Le Théoricien.
Jan 23 at 11:35
$begingroup$
It is just the usual one: $[A,B]=AB-BA$.
$endgroup$
– Le Théoricien.
Jan 23 at 11:35
add a comment |
1 Answer
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If $V$ and $W$ are tangent to N, it means that there are vector fields $v$ and $w$ in $mathfrak X(N)$ such that for any $xin N$ we have $V_{E(x)}=E_*v_x$ and the same is true for $W$. To be able to interpret things properly, assume that $V$ and $W$ are smoothly extended off $E(N)$.
Then $v$ and $V$ are $E$-related and so are $w$ and $W$.
But we know that for $E$-related vector fields the commutators are also $E$-related, so we have (restricted to $E(N)$) $$ [V,W]=E_*[v,w],$$
Implying that the commutator is tangent and is independent of the extensions.
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
$endgroup$
$begingroup$
Thanks for the comment! But where do we use the fact that $mathfrak{X}(M)ni V,W$?
$endgroup$
– Le Théoricien.
Jan 23 at 15:51
$begingroup$
Nevermind! That,s fine,
$endgroup$
– Le Théoricien.
Jan 23 at 16:00
add a comment |
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1 Answer
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$begingroup$
If $V$ and $W$ are tangent to N, it means that there are vector fields $v$ and $w$ in $mathfrak X(N)$ such that for any $xin N$ we have $V_{E(x)}=E_*v_x$ and the same is true for $W$. To be able to interpret things properly, assume that $V$ and $W$ are smoothly extended off $E(N)$.
Then $v$ and $V$ are $E$-related and so are $w$ and $W$.
But we know that for $E$-related vector fields the commutators are also $E$-related, so we have (restricted to $E(N)$) $$ [V,W]=E_*[v,w],$$
Implying that the commutator is tangent and is independent of the extensions.
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
$endgroup$
$begingroup$
Thanks for the comment! But where do we use the fact that $mathfrak{X}(M)ni V,W$?
$endgroup$
– Le Théoricien.
Jan 23 at 15:51
$begingroup$
Nevermind! That,s fine,
$endgroup$
– Le Théoricien.
Jan 23 at 16:00
add a comment |
$begingroup$
If $V$ and $W$ are tangent to N, it means that there are vector fields $v$ and $w$ in $mathfrak X(N)$ such that for any $xin N$ we have $V_{E(x)}=E_*v_x$ and the same is true for $W$. To be able to interpret things properly, assume that $V$ and $W$ are smoothly extended off $E(N)$.
Then $v$ and $V$ are $E$-related and so are $w$ and $W$.
But we know that for $E$-related vector fields the commutators are also $E$-related, so we have (restricted to $E(N)$) $$ [V,W]=E_*[v,w],$$
Implying that the commutator is tangent and is independent of the extensions.
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
$endgroup$
$begingroup$
Thanks for the comment! But where do we use the fact that $mathfrak{X}(M)ni V,W$?
$endgroup$
– Le Théoricien.
Jan 23 at 15:51
$begingroup$
Nevermind! That,s fine,
$endgroup$
– Le Théoricien.
Jan 23 at 16:00
add a comment |
$begingroup$
If $V$ and $W$ are tangent to N, it means that there are vector fields $v$ and $w$ in $mathfrak X(N)$ such that for any $xin N$ we have $V_{E(x)}=E_*v_x$ and the same is true for $W$. To be able to interpret things properly, assume that $V$ and $W$ are smoothly extended off $E(N)$.
Then $v$ and $V$ are $E$-related and so are $w$ and $W$.
But we know that for $E$-related vector fields the commutators are also $E$-related, so we have (restricted to $E(N)$) $$ [V,W]=E_*[v,w],$$
Implying that the commutator is tangent and is independent of the extensions.
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
$endgroup$
If $V$ and $W$ are tangent to N, it means that there are vector fields $v$ and $w$ in $mathfrak X(N)$ such that for any $xin N$ we have $V_{E(x)}=E_*v_x$ and the same is true for $W$. To be able to interpret things properly, assume that $V$ and $W$ are smoothly extended off $E(N)$.
Then $v$ and $V$ are $E$-related and so are $w$ and $W$.
But we know that for $E$-related vector fields the commutators are also $E$-related, so we have (restricted to $E(N)$) $$ [V,W]=E_*[v,w],$$
Implying that the commutator is tangent and is independent of the extensions.
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
answered Jan 23 at 12:54
Bence RacskóBence Racskó
3,393823
3,393823
$begingroup$
Thanks for the comment! But where do we use the fact that $mathfrak{X}(M)ni V,W$?
$endgroup$
– Le Théoricien.
Jan 23 at 15:51
$begingroup$
Nevermind! That,s fine,
$endgroup$
– Le Théoricien.
Jan 23 at 16:00
add a comment |
$begingroup$
Thanks for the comment! But where do we use the fact that $mathfrak{X}(M)ni V,W$?
$endgroup$
– Le Théoricien.
Jan 23 at 15:51
$begingroup$
Nevermind! That,s fine,
$endgroup$
– Le Théoricien.
Jan 23 at 16:00
$begingroup$
Thanks for the comment! But where do we use the fact that $mathfrak{X}(M)ni V,W$?
$endgroup$
– Le Théoricien.
Jan 23 at 15:51
$begingroup$
Thanks for the comment! But where do we use the fact that $mathfrak{X}(M)ni V,W$?
$endgroup$
– Le Théoricien.
Jan 23 at 15:51
$begingroup$
Nevermind! That,s fine,
$endgroup$
– Le Théoricien.
Jan 23 at 16:00
$begingroup$
Nevermind! That,s fine,
$endgroup$
– Le Théoricien.
Jan 23 at 16:00
add a comment |
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$begingroup$
There are a few possible approaches, depending on your definition of the commutator.
$endgroup$
– Amitai Yuval
Jan 23 at 6:49
$begingroup$
It is just the usual one: $[A,B]=AB-BA$.
$endgroup$
– Le Théoricien.
Jan 23 at 11:35