Geometry of Jacobi manifolds

It is helpful to give a very brief introduction to Jacobi manifolds in order to highlight what has been the guiding principle of my research activity so far.

As it is well-known symplectic manifolds provide the natural setting for the Hamiltonian formulation of the ordinary differential equations of classical mechanics, with the basic example of symplectic manifold being the cotangent bundle equipped with the canonical symplectic form. Contact manifolds play a fundamental rôle in developing the geometric theory of partial differential equations, with the canonical example of contact manifold being the first order jet bundle of a line bundle together with the Cartan distribution. Further, as a generalization of symplectic structures, the Poisson structures appear naturally in the description of the reduced dynamics of mechanical systems with symmetries (cf. for instance the KKS Poisson structure on the dual of a Lie algebra and the Euler-Arnol’d equations for Hamiltonian dynamics on Lie groups). Now Jacobi manifolds encompass, generalizing and unifying, all the preceding geometric structures: (non-necessarily coorientable) contact manifolds, locally conformal symplectic (lcs) manifolds, and Poisson manifolds (so, in particular, also symplectic manifolds). A Jacobi structure on a manifold M is given by a line bundle L → M equipped with a Jacobi bracket (or Jacobi bi-derivation), i.e. a Lie bracket on sections of L → M which is a first order differential operator in both entries. Jacobi manifolds were originally introduced by [Kirillov 1976] and independently by [Lichnerowicz 1978]. Notice that Kirillov’s approach to Jacobi manifolds via (non-necessarily trivial) line bundles will be systematically preferred to Lichnerowicz’s one via Jacobi pairs. Not only the former is slightly more general than the latter: it also allows us to take full advantage of the calculus of first-order multidifferential operators. In view of its great generality, Jacobi geometry provides a unified explanation both for the ubiquitous appearence of contact and symplectic geometry in many physical problems, and for the mirror-type correspondence existing between constructions/results in symplectic and contact geometry in the same spirit of the following quotation

"Every theorem in symplectic geometry may be formulated as a contact geometry theorem and

 an assertion in contact geometry can be translated in the language of symplectic geometry"

- Vladimir Igorevich Arnold

Nonetheless Jacobi manifolds are far less investigated than Poisson manifolds. Hence my research activity fits in the wider project aiming at filling the existing gap in our understanding of Jacobi manifolds.