Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 059, 22 pages      arXiv:1507.05405      https://doi.org/10.3842/SIGMA.2017.059

Remarks on Contact and Jacobi Geometry

Andrew James Bruce a, Katarzyna Grabowska b and Janusz Grabowski c
a) Mathematics Research Unit, University of Luxembourg, Luxembourg
b) Faculty of Physics, University of Warsaw, Poland
c) Institute of Mathematics, Polish Academy of Sciences, Poland

Received January 16, 2017, in final form July 17, 2017; Published online July 26, 2017

Abstract
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 linear Kirillov structures, 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 ${\rm GL}(1,{\mathbb R})$-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. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

Key words: symplectic structures; contact structures; Poisson structures; Jacobi structures; principal bundles; Lie groupoids; symplectic groupoids.

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