Geometry of coisotropic submanifolds in symplectic and K\"ahler manifolds

Abstract

The first purpose of this paper is to generalize the well-known Maslov indices of maps of open Riemann surfaces with boundary lying on Lagrangian submanifolds to maps with boundary lying on coisotropic submanifolds in symplectic manifolds. For this purpose, we first define the notion of {\it Maslov loops} of coisotropic Grassmanians and their indices. Then we introduce the notions of {\it transverse Maslov bundle} of coisotropic submanifolds, and {\it gradable coisotropic submanifolds}. We then define {\it graded coisotropic submanifolds} and the {\it coisotropic Maslov index} of the maps with boundary lying on such graded coisotropic submanifolds, which reduces to the standard Maslov index of disc maps for the case of Lagrangian submanifolds. The second purpose is to study the geometry of coisotropic submanifolds in K\"ahler manifolds. We introduce the notion of the leafwise mean curvature form and transverse canonical bundle of coisotropic submanifolds and study various geometric properties thereof. Finally we combine all these to define the notion of {\it special coisotropic submanifolds} for the case of Calabi-Yau manifolds, and prove various consequences on their properties of the coisotropic Maslov indices.