Md Rabiul Islam
Articles written in Proceedings – Mathematical Sciences
Volume 119 Issue 3 June 2009 pp 333-343
Smooth Maps of a Foliated Manifold in a Symplectic Manifold
Let 𝑀 be a smooth manifold with a regular foliation $\mathcal{F}$ and a 2-form 𝜔 which induces closed forms on the leaves of $\mathcal{F}$ in the leaf topology. A smooth map $f:(M,\mathcal{F})\longrightarrow(N, \sigma)$ in a symplectic manifold $(N, \sigma)$ is called a
If 𝑓 is a foliated symplectic immersion then the derivative map $Df$ gives rise to a bundle morphism $F:TM\longrightarrow TN$ which restricts to a monomorphism on $T\mathcal{F}\subseteq TM$ and satisfies the condition $F^∗\sigma=\omega$ on $T\mathcal{F}$. 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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