On contact groupoids and their symplectification (Q2771410)

The article continues publications of \textit{P. Libermann} [Diff. Geom. Appl., Proc. Conf. Opava 1992, 29-45 (1993; Zbl 0805.53034)] and \textit{C. Albert} and \textit{P. Dazord} [Math. Sci. Res. Inst. Publ. 20, 1-11 (1991; Zbl 0733.58019)]. The notion of contact groupoid \(\Gamma\) is generalized in the following way: The previous condition \(\varphi^* \omega=-\omega\) (where \(\varphi:x \mapsto x^{-1}\) is the symmetry of \(\Gamma)\) is replaced by \(\varphi^*\omega= -\omega/c\) (where \(c\) is a \(\mathbb{R}^+\)-valued function satisfying \(Ec=0\) for the Reeb vector field \(E\), and \(c(xy)= c(x)c(y)\) whenever \(xy\) is defined).NEWLINENEWLINENEWLINEResults: The symplectification \(\widetilde\Gamma\) is endowed with the induced symplectic groupoid structure if and only if \(\Gamma\) is a contact groupoid. The unit manifold \(\Gamma_0\) is a Legendrean submanifold of \(\Gamma\) endowed with a Jacobi structure. A contact groupoid \(\alpha,\beta: \Gamma\to \Gamma_0\) admits a definition sheaf composed of the germs of \(\omega\)-Hamiltonian vector field corresponding to the first integrals of the \(\alpha\)-foliation.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00036].