A Maslov Map for Coisotropic Submanifolds, Leaf-wise Fixed Points and Presymplectic Non-Embeddings

Abstract

Let $(M,\omega)$ be a symplectic manifold, $N\subseteq M$ a coisotropic submanifold, and $\Sigma$ a compact oriented (real) surface. The main purpose of this article is to associate a Maslov map to this data, which naturally generalizes the Lagrangian Maslov index, and twice the first Chern number. This map assigns a real number to every suitable homotopy class of maps $\Sigma\to M$ that send each connected component of $\partial\Si$ to an isotropic leaf of $N$. In the case $\codim N<\dim M/2$, the idea is to use the linear holonomy of the isotropic foliation of $N$ to compensate for the loss of boundary data. Some properties of the index and examples are discussed.