2003/10/31 by Yong-Geun Oh, Oh, Yong-Geun
Mathematics · #32J40 #53D05 #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:32J40 #msc:53D05
paper · pdf · doi:10.48550/arxiv.math/0310482
30 pages, Definitions of coisotropic Maslov indices and special coisotropic submanifolds are corrected. Notions of gradable and graded coisotropic manifolds are added. Many improvements in presentation
arxiv created 2003/12/10 · arxiv updated 2009/12/01
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 Kähler 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.