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.


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