Mixing
Frobenius Theorem for $p=2$
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
The theorem states that if a $c$ distribution $D$ is invlutive on $M^d$ then there is a only one integral submanifold $N$ and local charts $(z_1,...z_n)$ such that $N$ is the slice where $z_i=const$, $i=c+1,...,d$.
So for $p=2$ I choose a neighborhood $U$ of $p$ and write $D$ as $span<X_1,X_2>$, where $X_i$ are smooths vector field on $U$ and then use a lemma that states that i can found a neighborhood $V subset U$ and local charts $(y_1,...,y_d)$ such that $X_1=partial/partial y_1$.
so i write a new vector field
$Y_1=x_1$
$Y_2=X_2-Y_2(y_1)Y_1$
and as $Y_2$ doesn't depend on $y_1$ ($Y_2(y_1)=0$), and so i can afirm that $Y_2^'$ spans a 1-dimensional distribuition on the slice S where $y_1=const$.
Applying the Lemma used before on $Y_2$ we find a new coordinates chart of a neighborhod of $p$
$(A,x_1,...,x_n)$ such that $Y_2=partial/ partial x_2$ on $A$
My doubt is that if I can say on $Acap U$ $partial y_1/partial x_i=0$ if $i=2,...n$ and so $X_1=partial/partial x_1$ and the theorem is proved.
If not, how should i proceed?
differential-geometry manifolds vector-bundles
asked Jul 26 at 5:08
Eduardo Silva
60039
add a comment |Â
up vote
0
down vote
favorite
The theorem states that if a $c$ distribution $D$ is invlutive on $M^d$ then there is a only one integral submanifold $N$ and local charts $(z_1,...z_n)$ such that $N$ is the slice where $z_i=const$, $i=c+1,...,d$.
So for $p=2$ I choose a neighborhood $U$ of $p$ and write $D$ as $span<X_1,X_2>$, where $X_i$ are smooths vector field on $U$ and then use a lemma that states that i can found a neighborhood $V subset U$ and local charts $(y_1,...,y_d)$ such that $X_1=partial/partial y_1$.
so i write a new vector field
$Y_1=x_1$
$Y_2=X_2-Y_2(y_1)Y_1$
and as $Y_2$ doesn't depend on $y_1$ ($Y_2(y_1)=0$), and so i can afirm that $Y_2^'$ spans a 1-dimensional distribuition on the slice S where $y_1=const$.
Applying the Lemma used before on $Y_2$ we find a new coordinates chart of a neighborhod of $p$
$(A,x_1,...,x_n)$ such that $Y_2=partial/ partial x_2$ on $A$
My doubt is that if I can say on $Acap U$ $partial y_1/partial x_i=0$ if $i=2,...n$ and so $X_1=partial/partial x_1$ and the theorem is proved.
If not, how should i proceed?
differential-geometry manifolds vector-bundles
asked Jul 26 at 5:08
Eduardo Silva
60039
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The theorem states that if a $c$ distribution $D$ is invlutive on $M^d$ then there is a only one integral submanifold $N$ and local charts $(z_1,...z_n)$ such that $N$ is the slice where $z_i=const$, $i=c+1,...,d$.
So for $p=2$ I choose a neighborhood $U$ of $p$ and write $D$ as $span<X_1,X_2>$, where $X_i$ are smooths vector field on $U$ and then use a lemma that states that i can found a neighborhood $V subset U$ and local charts $(y_1,...,y_d)$ such that $X_1=partial/partial y_1$.
so i write a new vector field
$Y_1=x_1$
$Y_2=X_2-Y_2(y_1)Y_1$
and as $Y_2$ doesn't depend on $y_1$ ($Y_2(y_1)=0$), and so i can afirm that $Y_2^'$ spans a 1-dimensional distribuition on the slice S where $y_1=const$.
Applying the Lemma used before on $Y_2$ we find a new coordinates chart of a neighborhod of $p$
$(A,x_1,...,x_n)$ such that $Y_2=partial/ partial x_2$ on $A$
My doubt is that if I can say on $Acap U$ $partial y_1/partial x_i=0$ if $i=2,...n$ and so $X_1=partial/partial x_1$ and the theorem is proved.
If not, how should i proceed?
differential-geometry manifolds vector-bundles
asked Jul 26 at 5:08
Eduardo Silva
60039
The theorem states that if a $c$ distribution $D$ is invlutive on $M^d$ then there is a only one integral submanifold $N$ and local charts $(z_1,...z_n)$ such that $N$ is the slice where $z_i=const$, $i=c+1,...,d$.
So for $p=2$ I choose a neighborhood $U$ of $p$ and write $D$ as $span<X_1,X_2>$, where $X_i$ are smooths vector field on $U$ and then use a lemma that states that i can found a neighborhood $V subset U$ and local charts $(y_1,...,y_d)$ such that $X_1=partial/partial y_1$.
so i write a new vector field
$Y_1=x_1$
$Y_2=X_2-Y_2(y_1)Y_1$
and as $Y_2$ doesn't depend on $y_1$ ($Y_2(y_1)=0$), and so i can afirm that $Y_2^'$ spans a 1-dimensional distribuition on the slice S where $y_1=const$.
Applying the Lemma used before on $Y_2$ we find a new coordinates chart of a neighborhod of $p$
$(A,x_1,...,x_n)$ such that $Y_2=partial/ partial x_2$ on $A$
My doubt is that if I can say on $Acap U$ $partial y_1/partial x_i=0$ if $i=2,...n$ and so $X_1=partial/partial x_1$ and the theorem is proved.
If not, how should i proceed?
differential-geometry manifolds vector-bundles
asked Jul 26 at 5:08
Eduardo Silva
60039
asked Jul 26 at 5:08
Eduardo Silva
60039
asked Jul 26 at 5:08
Eduardo Silva
60039
asked Jul 26 at 5:08
Eduardo Silva
60039
60039
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2863090%2ffrobenius-theorem-for-p-2%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password