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edited Apr 29 2011 at 14:54 |
Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.
0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$.
(Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)
1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$.
For any $p\in M$, there exists a lagrangian submanifold of $(M,\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.
Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}}$,
the restriction of $T_x \mathcal{F}x=\textnormal{span}\left(\frac{\partial}{\partial\phi{n+1}}|x,\ldots,\frac{\partial}{\partial\phi{2n}}|x\right)$, for any \mathcal{F}$ on $x\in U$ ,is generated by $\frac{\partial}{\partial\phi_{n+1}},\ldots,\frac{\partial}{\partial\phi_{2n}}$,
and consequently $\phi{n+1}=\ldots=\phi_{2n}=0$ \phi_{n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.
This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.
2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^{\ast}:T^{\ast}L\to L$,
$f_{\ast}\omega$ is the canonical symplectic on $T^{\ast}L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^{\ast}_L$.
This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.
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edited Apr 29 2011 at 14:47 |
Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.
0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$.
(Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)
1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$.
For any $p\in M$, there exists a lagrangian submanifold of $(M,\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.
Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}}$,
$T_x \mathcal{F}x=\frac{\partial}{\partial\phix=\textnormal{span}\left(\frac{\partial}{\partial\phi{n+1}}|x,\ldots,\frac{\partial}{\partial\phi{2n}}|x$x\right)$, for any $x\in U$,
and consequently $\phi{n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.
This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.
2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^{\ast}:T^{\ast}L\to L$,
$f_{\ast}\omega$ is the canonical symplectic on $T^{\ast}L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^{\ast}_L$.
This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.
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edited Apr 29 2011 at 14:42 |
Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.
0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$.
(Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)
1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$.
For any $p\in M$, there exists a lagrangian submanifold of $(M,\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.
Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}}$,
$T_x\mathcal{F}x=\textrm{span}\left(\frac{\partial}{\partial\phiT_x \mathcal{F}x=\frac{\partial}{\partial\phi{n+1}}|x,\ldots,\frac{\partial}{\partial\phi{2n}}|x\right)$x$, for any $x\in U$,
and consequently $\phi{n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.
This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.
2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^{\ast}:T^{\ast}L\to L$,
$f_{\ast}\omega$ is the canonical symplectic on $T^{\ast}L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^{\ast}_L$.
This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.
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edited Apr 29 2011 at 14:34 |
Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.
0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$.
(Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)
1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$.
For any $p\in M$, there exists a lagrangian submanifold of $(M\omega)$ (M,\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.
Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}$d\phi_{n+i}}$,
$T_x\mathcal{F}x=\textrm{span}(\frac{\partial}{\partial\phix=\textrm{span}\left(\frac{\partial}{\partial\phi{n+1}|n+1}}|x,\ldots,\frac{\partial}{\partial\phi{2n}|x)$2n}}|x\right)$, for any $x\in U$,
and consequently $\phi{n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.
This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.
2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^{\ast}:T^{\ast}L\to L$,
$f_{\ast}\omega$ is the canonical symplectic on $T^{\ast}L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^{\ast}_L$.
This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.
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edited Apr 29 2011 at 14:28 |
Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.
0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$.
(Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)
1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$.
For any $p\in M$, there exists a lagrangian submanifold of $(M\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.
Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega=\sum_{i=1}^n{d\phi_i\wedge \omega= \sum_{i=1}^{n}{d\phi_i \wedge d\phi_{n+i}$,
$T_x\mathcal{F}x=\textrm{span}(\frac{\partial}{\partial\phi{n+1}|x,\ldots,\frac{\partial}{\partial\phi{2n}|x)$, for any $x\in U$,
and consequently $\phi{n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.
This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.
2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^*:T^L\to \tau_L^{\ast}:T^{\ast}L\to L$,
$f_\omega$ f_{\ast}\omega$ is the canonical symplectic on T^L$,$T^{\ast}L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^_L$.\tau^{\ast}_L$.
This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.
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answered Apr 29 2011 at 14:22 |
Dear Theo Johnson-Freyd, I hope to have at least partially understood the content of your question, and that my answer could be useful.
0.Setting and specification of the terminology.
In a symplectic $2n$-dimensional manifold $(M,\omega)$, let be given a lagrangian foliation $\mathcal{F}$, i.e. a foliation of $M$ whose leaves are lagrangian w.r.t. $\omega$.
(Instead, I mean a lagrangian fibration of $(M,\omega)$ as a surjective summersion $f:M\to B$ whose fibers are lagrangian w.r.t. $\omega$. Any fibration determines a foliation but the converse is not true. The difference will be immaterial in my point(1), but not so in my point(2).)
1.Local Existence of lagrangian submanifolds transversal to $\mathcal{F}$.
For any $p\in M$, there exists a lagrangian submanifold of $(M\omega)$ which passes through $p$ and is transversal to $\mathcal{F}$.
Infact, for any $p\in M$, there exists a chart $(U,\phi)$ for $M$ centered at $p$, such that:
$\omega=\sum_{i=1}^n{d\phi_i\wedge d\phi_{n+i}$,
$T_x\mathcal{F}x=\textrm{span}(\frac{\partial}{\partial\phi{n+1}|x,\ldots,\frac{\partial}{\partial\phi{2n}|x)$, for any $x\in U$,
and consequently $\phi{n+1}=\ldots=\phi_{2n}=0$ is a local lagrangian submanifold of $(M,\omega)$ passing through $p$ and transverval to $\mathcal{F}$.
This is just the Caratheodory-Jacobi-Lie theorem, applied starting with a system $d\phi_1,\ldots,d\phi_n$ of $1$-forms which locally generates the distribution corresponding to the lagrangian foliation $\mathcal{F}$.
2. A relative globalization.
If $L$, a lagrangian submanifold of $(M,\omega)$, is transversal to $\mathcal{F}$, then there exists a diffeomorphism $f$ from an open neigborhood of $L$ in $M$ onto an open set in $T^*L$ such that:
$f|_L$ is the zero section of $\tau_L^*:T^L\to L$,
$f_\omega$ is the canonical symplectic on T^L$,
and $f$ takes the leaves of $\mathcal{F}$ in the fibers of $\tau^_L$.
This is just Theorem 7.1 in "Symplectic Manifolds and their Lagrangian submanifolds" of A.Weinstein.
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