Let @math be a symplectic manifold, @math a coisotropic submanifold, and @math a compact oriented (real) surface. I define a natural Maslov index for each continuous map @math that sends every connected component of @math to some isotropic leaf of @math . This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of @math to compensate for the loss of boundary data in the case codimension @math . The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplectic automorphisms. I prove a lower bound on the number of leafwise fixed points of a Hamiltonian diffeomorphism, if @math is geometrically bounded and @math is closed, regular (i.e. "fibering"), and monotone. As an application, we obtain a presymplectic non-embedding result. I also prove a coisotropic version of the Audin conjecture.
@cite_11: We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.
Assume that the action of @math on @math is free. Then by Lemma below @math is a closed, monotone regular coisotropic submanifold. Hence if @math is non-zero for some @math then it follows from Theorem that @math is not leafwise displaceable (and hence not displaceable). Thus in this case we obtain a stronger statement than in Theorem 1.7 in @cite_11 , provided that also @math .