## Smooth maps of a foliated manifold in a symplectic manifold.(English)Zbl 1173.53309

Summary: Let $$M$$ be a smooth manifold with a regular foliation $${\mathcal F}$$ and a 2-form $$\omega$$ which induces closed forms on the leaves of $${\mathcal F}$$ in the leaf topology. A smooth map $$f: (M,{\mathcal F})\rightarrow (N,\sigma)$$ in a symplectic manifold $$(N,\sigma)$$ is called a foliated symplectic immersion if $$f$$ restricts to an immersion on each leaf of the foliation and further, the restriction of $$f^*\sigma$$ is the same as the restriction of $$\omega$$ on each leaf of the foliation. If $$f$$ is a foliated symplectic immersion then the derivative map $$Df$$ gives rise to a bundle morphism $$F: TM\rightarrow TN$$ 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.

### MSC:

 53C12 Foliations (differential geometric aspects) 53D05 Symplectic manifolds (general theory)
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### References:

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