On the rigidity of the coisotropic Maslov index on certain rational symplectic manifolds

TL;DR

This paper investigates the Maslov index for loops in coisotropic submanifolds, proving its rigidity for stable cases and establishing a nearby existence theorem within certain rational symplectic manifolds.

Contribution

It introduces a new perspective on the Maslov index for coisotropic submanifolds and proves rigidity and existence results in broad classes of symplectic manifolds.

Findings

Rigidity of the Maslov index for stable coisotropic submanifolds

Existence of nearby loops with prescribed Maslov index

Applicability to a broad class of rational symplectic manifolds

Abstract

We revisit the definition of the Maslov index of loops in coisotropic submanifolds tangent to the characteristic foliation of this submanifold. This Maslov index is given by the mean index of a certain symplectic path which is a lift of the holonomy along the loop. We prove a result on the rigidity of the Maslov index for stable coisotropic submanifolds in a broad class of ambient symplectic manifolds. Furthermore, we establish a nearby existence theorem for the same class of ambient manifolds.