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Every maximal isotropic subbundle $Lsubset TMoplus T^*M$ can be express as $L(E,alpha)$
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Every maximal isotropic (totally null) subspaces $Lsubset Voplus V^*$ (with respect to the natural pairing $langle X+xi,Y+etarangle = eta(X) + xi(Y)$) can be express as $L(E,alpha)$ for some appropriate $E⊆V$ and 2-form $α∈Λ^2(E)$:
$$ L(E,alpha) = X+xiin Eoplus V^* : xi . tag1 $$
I already asked a question about the proof of this fact here.
Now, I'm wondering if every maximal isotropic subbundle $Lsubset TMoplus T^*M$ can also be expressed in the same way. Namely, if given $L$, there exists a subbundle $Esubset TM$ and 2-form $alphainGamma^infty(Lambda^2E)$ such that
$$L=(p,X)+(p,xi)in Eoplus T^*M : mboxfor each pin M ;; xi . $$
According to Gualtieri, this identification is possible only in regular points $pin M$, in the sense that the leaf dimension is constant in a neighbourhood
$U$ of $p$. But I can't figure why leaves are important here. To prove my claim I thought of using the vector bundle construction theorem using as typical fibe the vector space of (1), where in this case $E$ would be the typical fibre of the vector subbundle $E$.
What do you think?
Context. This question arises in the proof of the Generalized Darboux theorem (Gualtieri's thesis pp.56--57). To prove the theorem he uses a previous proposition 3.12 about transverse folitations. Then, he must to restrict the Darboux theorem to regular points of the manifold. However, I think that a maximal isotropic subbundle can be expressed in the above form, independently the regularity of the point of the manifold but he asys We saw in Proposition 4.19 that in a regular neighbourhood, a generalized complex structure may
be expressed as $L(E, ε)$ where $E < T ⊗ C$ is an involutive subbundle and $ε ∈ C^∞(∧^2E^∗)$ satisfies $d_Eε = 0$. Probably he is right but why do we need regularity?
PD: Feel free to add more tags. I have added only one because there isn't any about generalized geometry.
differential-geometry
asked Jul 27 at 7:27
Dog_69
3301320
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Every maximal isotropic (totally null) subspaces $Lsubset Voplus V^*$ (with respect to the natural pairing $langle X+xi,Y+etarangle = eta(X) + xi(Y)$) can be express as $L(E,alpha)$ for some appropriate $E⊆V$ and 2-form $α∈Λ^2(E)$:
$$ L(E,alpha) = X+xiin Eoplus V^* : xi . tag1 $$
I already asked a question about the proof of this fact here.
Now, I'm wondering if every maximal isotropic subbundle $Lsubset TMoplus T^*M$ can also be expressed in the same way. Namely, if given $L$, there exists a subbundle $Esubset TM$ and 2-form $alphainGamma^infty(Lambda^2E)$ such that
$$L=(p,X)+(p,xi)in Eoplus T^*M : mboxfor each pin M ;; xi . $$
According to Gualtieri, this identification is possible only in regular points $pin M$, in the sense that the leaf dimension is constant in a neighbourhood
$U$ of $p$. But I can't figure why leaves are important here. To prove my claim I thought of using the vector bundle construction theorem using as typical fibe the vector space of (1), where in this case $E$ would be the typical fibre of the vector subbundle $E$.
What do you think?
Context. This question arises in the proof of the Generalized Darboux theorem (Gualtieri's thesis pp.56--57). To prove the theorem he uses a previous proposition 3.12 about transverse folitations. Then, he must to restrict the Darboux theorem to regular points of the manifold. However, I think that a maximal isotropic subbundle can be expressed in the above form, independently the regularity of the point of the manifold but he asys We saw in Proposition 4.19 that in a regular neighbourhood, a generalized complex structure may
be expressed as $L(E, ε)$ where $E < T ⊗ C$ is an involutive subbundle and $ε ∈ C^∞(∧^2E^∗)$ satisfies $d_Eε = 0$. Probably he is right but why do we need regularity?
PD: Feel free to add more tags. I have added only one because there isn't any about generalized geometry.
differential-geometry
asked Jul 27 at 7:27
Dog_69
3301320
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Every maximal isotropic (totally null) subspaces $Lsubset Voplus V^*$ (with respect to the natural pairing $langle X+xi,Y+etarangle = eta(X) + xi(Y)$) can be express as $L(E,alpha)$ for some appropriate $E⊆V$ and 2-form $α∈Λ^2(E)$:
$$ L(E,alpha) = X+xiin Eoplus V^* : xi . tag1 $$
I already asked a question about the proof of this fact here.
Now, I'm wondering if every maximal isotropic subbundle $Lsubset TMoplus T^*M$ can also be expressed in the same way. Namely, if given $L$, there exists a subbundle $Esubset TM$ and 2-form $alphainGamma^infty(Lambda^2E)$ such that
$$L=(p,X)+(p,xi)in Eoplus T^*M : mboxfor each pin M ;; xi . $$
According to Gualtieri, this identification is possible only in regular points $pin M$, in the sense that the leaf dimension is constant in a neighbourhood
$U$ of $p$. But I can't figure why leaves are important here. To prove my claim I thought of using the vector bundle construction theorem using as typical fibe the vector space of (1), where in this case $E$ would be the typical fibre of the vector subbundle $E$.
What do you think?
Context. This question arises in the proof of the Generalized Darboux theorem (Gualtieri's thesis pp.56--57). To prove the theorem he uses a previous proposition 3.12 about transverse folitations. Then, he must to restrict the Darboux theorem to regular points of the manifold. However, I think that a maximal isotropic subbundle can be expressed in the above form, independently the regularity of the point of the manifold but he asys We saw in Proposition 4.19 that in a regular neighbourhood, a generalized complex structure may
be expressed as $L(E, ε)$ where $E < T ⊗ C$ is an involutive subbundle and $ε ∈ C^∞(∧^2E^∗)$ satisfies $d_Eε = 0$. Probably he is right but why do we need regularity?
PD: Feel free to add more tags. I have added only one because there isn't any about generalized geometry.
differential-geometry
asked Jul 27 at 7:27
Dog_69
3301320
Every maximal isotropic (totally null) subspaces $Lsubset Voplus V^*$ (with respect to the natural pairing $langle X+xi,Y+etarangle = eta(X) + xi(Y)$) can be express as $L(E,alpha)$ for some appropriate $E⊆V$ and 2-form $α∈Λ^2(E)$:
$$ L(E,alpha) = X+xiin Eoplus V^* : xi . tag1 $$
I already asked a question about the proof of this fact here.
Now, I'm wondering if every maximal isotropic subbundle $Lsubset TMoplus T^*M$ can also be expressed in the same way. Namely, if given $L$, there exists a subbundle $Esubset TM$ and 2-form $alphainGamma^infty(Lambda^2E)$ such that
$$L=(p,X)+(p,xi)in Eoplus T^*M : mboxfor each pin M ;; xi . $$
According to Gualtieri, this identification is possible only in regular points $pin M$, in the sense that the leaf dimension is constant in a neighbourhood
$U$ of $p$. But I can't figure why leaves are important here. To prove my claim I thought of using the vector bundle construction theorem using as typical fibe the vector space of (1), where in this case $E$ would be the typical fibre of the vector subbundle $E$.
What do you think?
Context. This question arises in the proof of the Generalized Darboux theorem (Gualtieri's thesis pp.56--57). To prove the theorem he uses a previous proposition 3.12 about transverse folitations. Then, he must to restrict the Darboux theorem to regular points of the manifold. However, I think that a maximal isotropic subbundle can be expressed in the above form, independently the regularity of the point of the manifold but he asys We saw in Proposition 4.19 that in a regular neighbourhood, a generalized complex structure may
be expressed as $L(E, ε)$ where $E < T ⊗ C$ is an involutive subbundle and $ε ∈ C^∞(∧^2E^∗)$ satisfies $d_Eε = 0$. Probably he is right but why do we need regularity?
PD: Feel free to add more tags. I have added only one because there isn't any about generalized geometry.
differential-geometry
asked Jul 27 at 7:27
Dog_69
3301320
asked Jul 27 at 7:27
Dog_69
3301320
asked Jul 27 at 7:27
Dog_69
3301320
asked Jul 27 at 7:27
Dog_69
3301320
3301320
add a comment |Â
add a comment |Â
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