Fibrés symplectiques et la géométrie des difféomorphismes hamiltoniens
This thesis presents a reasonably complete account of the elements theory of symplectic and Hamiltonian fibrations.  We assume a familiarity and comfort with the basic notions of differential geometry and algebraic topology but little else.  Proceeding from this, the first chapter develops the necessary notions from the theory of fiber bundles and G-fiber bundles, while the second chapter develops all the notions and theorems required to understand the later theory of symplectic fibrations.  Most notably the second chapter includes a detailed account of the classical relationship between the flux homomorphism and Hamiltonian isotopies.  The third chapter is where we develop the theory of symplectic and locally Hamiltonian fiber bundles, and in particular give an invariant construction of the coupling form on a symplectic fibration admitting an extension class.  the third chapter ends with a proof of a structure theorem characterizing those symplectic fibrations for which the structure group reduces to the Hamiltonian group.  In the final chapter, we present some applications of the theory of Hamiltonian fibrations by the way of characterizing the positive part of the Hofer norm of a Hamiltonian loop as the K-area of its associated Hamiltonian bundle over the sphere, and we finish by giving a proof of the non-degeneracy of the Hofer norm for closed symplectic manifolds.
G-fiber bundles Symplectic fiber bundles Hamiltonian fiber bundles Flux morphism Hofer norm K-area Coupling form