In this paper we give a homotopy classification of symplectic isometric immersions following Gromov’sh-principle theorem.

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 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 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 $F:TM\longrightarrow TN$ 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.

We give a homotopy classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient space. The result is an extension of Haefliger’s classification of foliations on open manifold in the contact setting. While proving the main theorem, we also prove a result on equidimensional isocontactimmersions on open contact manifolds.