## 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.