Some invariants of locally conformally symplectic structures (Q2708167)

A locally conformal symplectic structure \({\mathcal S}\) on a manifold \(M\) is an equivalence class of non-degenerate (conformally equivalent) 2-forms \(\Omega\), such that \(d\Omega=-\omega\wedge\Omega\) for some closed 1-form \(\omega\), which is called the Lee form of the structure. Infinitesimal automorphisms of \({\mathcal S}\) are vector fields \(X\) such that \(L_X\Omega= \delta_X \Omega\) for some function \(\delta_X\). Assuming the manifold is simply connected, \(\omega(X)+\delta_X\) is a constant \(l(X)\) and the map \(l:X\mapsto l(X)\) is a Lie algebra homomorphism, called the extended Lee homomorphism. NEWLINENEWLINENEWLINEThe main theorem of this note is about the integration of the exponential of \(l\), which gives rise to a homomorphism \({\mathcal L}\) on the group of automorphisms of the structure \({\mathcal S}\). A further result is about the analogues, for locally conformal symplectic geometry, of the Calabi homomorphisms in symplectic geometry. As an application, the author introduces the notion of Hamilton equations on the Galois covering of \(M\) associated to the Lee form \(\omega\), and uses this to arrive at a global formula for a Jacobi bracket on \(M\). NEWLINENEWLINENEWLINEProofs can be found in J. Geom. Phys. 39, 30-44 (2001; Zbl 1032.53017).