The Geometry of Partial Order on Contact Transformations of Prequantization Manifolds

Summary

In this chapter we find a connection between the Hofer metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in (12), of the quantomorphisms group of its prequantization manifold. This gives two main results: First, we calculate, partly, the geometry of the quantomorphism group of a prequantization manifold of an integral symplectic manifold which admits a certain Lagrangian foliation. Second, for every prequantization manifold we give a formula for the distance between a point and a distinguished curve in the metric space associated to its group of quantomorphisms. Moreover, our first result is a full computation of the geometry related to the symplectic linear group which can be considered as a subgroup of the contactomorphism group of suitable prequantization manifolds of complex projective space. In the course of the proof we use in an essential way the Maslov quasimorphism.