Remarks on contact and Jacobi geometry

We present an approach to Jacobi and contact geometry that makes many facts, presented in the
literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov
manifolds and Kirillov algebroids, i.e. homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL(1, R)-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. In this sense, the properly understood concept of a Jacobi structure is a specialisation rather than a generalisation of a Poission structure. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, as well as give new insight in the theory. For instance, we describe the structure of Lie groupoids with a compatible principal G-bundle structure and the ‘integrating objects’ for Kirillov algebroids, define canonical contact groupoids, and show that any contact groupoid has a canonical realisation as a contact subgroupoid of the latter.