Smooth maps of a foliated manifold in a symplectic manifold

Abstract

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, \( \mathcal{F} \)) → (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 \( \mathcal{F} \) ⊆ T M and satisfies the condition F*σ = ω 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.