Smooth maps of a foliated manifold in a symplectic manifold

TL;DR

This paper extends Gromov's $h$-principle for immersions of manifolds into symplectic manifolds to the foliated setting, showing that certain smooth maps are dense among continuous maps respecting cohomology classes.

Contribution

It proves a foliated version of Gromov's $h$-principle for smooth maps into symplectic manifolds, generalizing previous results to foliated manifolds.

Findings

The $C^0$-density of immersions satisfying cohomological conditions.

Extension of Gromov's $h$-principle to foliated manifolds.

Establishment of a foliated $h$-principle in symplectic geometry.

Abstract

The immersions of a smooth manifold $M$ in a symplectic manifold $(N,\sigma)$ inducing a given closed form $\omega$ on $M$ satisfy the $C^0$-dense $h$-principle in the space of all continuous maps which pull back the deRham cohomology class of $\sigma$ onto that of $\omega$. In this paper we prove a foliated version of this result due to Gromov.