Introduction to contact complete integrability

Abstract

Integrable Hamiltonian systems on symplectic manifolds have been well-studied. However, an intrinsic property of these kind of systems is that they can only live on even dimensional manifolds. To introduce a similar notion of integrability on odd dimensional manifolds, one needs another type of geometry: contact geometry. These lecture notes, aimed towards graduate students, serve as an introduction to this form of integrabilty. We will start by introducing the necessary concepts from contact geometry, which is the sister geometry of symplectic geometry. In the third chapter, we discuss the contact Hamiltonian vector field (which is similar to the standard Hamiltonian vector field) and the Jacobi bracket (which has similar dynamical properties as the Poisson bracket in the symplectic world). After this, we are able to introduce contact complete integrability: there exist (at least) two different notions of such integrability in the literature: one by Khesin & Tabachnikov (which has a more geometric nature) and one by Jovanovic & Jovanovic. We will show that both notions coincide. Subsequently, we will give an overview of the semi-local aspects of contact complete integrability, namely the behaviour of the dynamics near regular fibres (an Arnold-Liouville-like theorem by Jovanovic) and singular fibres (a local normal form as described by Miranda). Finally, we will introduce contact toric G-manifolds, which where classified by Lerman in 2003 (similar to the Delzant-classification of toric integrable systems). Throughout these notes, we will focus on the link with integrable Hamiltonian systems in symplectic geometry.