Geodesic flows and contact toric manifolds

Forward These notes are based on five 1.5 hour lectures on torus actions on contact mani-folds delivered at the summer school on Symplectic Geometry of Integrable Hamil-tonian Systems at Centre de Recerca Matemàtica in Barcelona in July 2001. Naturally the notes contain more material that could have been delivered in 7.5 hours. I am grateful to Carlos Curràs-Bosch and Eva Miranda, the organizers of the summer school, for their kind invitation to teach a course. Thanks are also due to the staff of the CRM without whom the summer school would not have been a success. The main theme of these notes is the topological study of contact toric man-ifolds, a relatively new class of manifolds that I find very interesting. A motivation for studying these manifolds comes from completely integrable systems {f 1 ,. .. , f n } on punctured cotangent bundles where each function f i is homogeneous of degree 1 (one can think of f i 's as symbols of first order pseudo-differential operators, but this is not essential). A punctured cotangent bundle is a symplec-tic cone whose base is naturally a contact manifold (this is explained in detail in Chapter 2). This observation leads to studying completely integrable systems on contact manifolds, whatever those are. The simplest (symplectic) completely integrable systems are the ones with global action-angle coordinates. The next simplest case is that of Hamiltonian torus actions. If the phase space is compact one ends up with (compact) symplectic toric manifolds. This is the theme of Ana Cannas's lectures delivered at the summer school. The corresponding case in the contact category is that of compact toric manifolds. We will use the excuse of studying completely integrable geodesic flows with homogeneous integrals to introduce various ideas essential for the classification of contact and symplectic toric manifolds. More specifically we will discuss in these notes contact moment maps, slices for group actions, sheaves andČech cohomol-ogy, orbifolds and Morse theory on orbifolds.