Smooth maps of a foliated manifold in a symplectic manifold |
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Authors: | Mahuya Datta Md Rabiul Islam |
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Institution: | 1.Statistics and Mathematics Unit,Indian Statistical Institute,Kolkata,India;2.Department of Pure Mathematics, University College of Science,University of Calcutta,Kolkata,India |
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Abstract: | Let M be a smooth manifold with a regular foliation $
\mathcal{F}
$
\mathcal{F}
and a 2-form ω which induces closed forms on the leaves of $
\mathcal{F}
$
\mathcal{F}
in the leaf topology. A smooth map f: (M, $
\mathcal{F}
$
\mathcal{F}
) → (N, σ) in a symplectic manifold (N, σ) is called a foliated symplectic immersion if f restricts to an immersion on each leaf of the foliation and further, the restriction of f*σ is the same as the restriction of ω on each leaf of the foliation.
If f is a foliated symplectic immersion then the derivative map Df gives rise to a bundle morphism F: TM → T N which restricts to a monomorphism on T
$
\mathcal{F}
$
\mathcal{F}
⊆ T M and satisfies the condition F*σ = ω on T
$
\mathcal{F}
$
\mathcal{F}
. A natural question is whether the existence of such a bundle map F ensures the existence of a foliated symplectic immersion f. As we shall see in this paper, the obstruction to the existence of such an f is only topological in nature. The result is proved using the h-principle theory of Gromov. |
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Keywords: | |
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