Integral affine geometry of Lagrangian bundles

Abstract

In this thesis, a bundle F →(M,ω) → B is said to be Lagrangian if (M,ω) is a 2n- dimensional symplectic manifold and the fibres are compact and connected Lagrangian submanifolds of (M,ω), i.e. ω |F = 0 for all F. This condition implies that the fibres and the base space are n-dimensional. Such bundles arise naturally in the study of a special class of dynamical systems in Hamiltonian mechanics, namely those called completely integrable Hamiltonian systems. A celebrated theorem due to Liouville [39], Mineur [46] and Arnol`d [2] provides a semi-global (i.e. in the neighbourhood of a fibre) symplectic classification of Lagrangian bundles, given by the existence of local action-angle coordinates. A proof of this theorem, due to Markus and Meyer [41] and Duistermaat [20], shows that the fibres and base space of a Lagrangian bundle are naturally integral affine manifolds, i.e. they admit atlases whose changes of coordinates can be extended to affine transformations of Rn which preserve the standard cocompact lattice Zn Rn. This thesis studies the problem of constructing Lagrangian bundles from the point of view of affinely at geometry. The first step to study this question is to construct topological universal Lagrangian bundles using the affine structure on the fibres. These bundles classify Lagrangian bundles topologically in the sense that every such bundle arises as the pullback of one universal bundle. However, not all bundles which are isomorphic to the pullback of a topological universal Lagrangian bundle are Lagrangian, as there exist further smooth and symplectic invariants. Even for bundles which admit local action-angle coordinates (these are classified up to isomorphism by topological universal Lagrangian bundles), there is a cohomological obstruction to the existence of an appropriate symplectic form on the total space, which has been studied by Dazord and Delzant in [18]. Such bundles are called almost Lagrangian. The second half of this thesis constructs the obstruction of Dazord and Delzant using the spectral sequence of a topological universal Lagrangian bundle. Moreover, this obstruction is shown to be related to a cohomological invariant associated to the integral affine geometry of the base space, called the radiance obstruction. In particular, it is shown that the integral a ne geometry of the base space of an almost Lagrangian bundle determines whether the bundle is, in fact, Lagrangian. New examples of (almost) Lagrangian bundles are provided to illustrate the theory developed.