Symplectic 4-manifolds via Lorentzian geometry (Q2832837)

In the paper, an interesting relation between Lorentzian geometry in dimension \(4\) and symplectic geometry is established: a complete null vector field \(k\) with geodesic flow, satisfying \({\mathrm{Ric}}(k,k)>0\), and such that there exists a smooth function \(f\) with \(k(f)\) nowhere vanishing, gives rise to a symplectic form and a Liouville vector field. Null surfaces along which \(k\) is tangent are Lagrangian submanifolds. It is further shown that the Ricci condition is not necessary, but completeness is necessary.NEWLINENEWLINEUsing the same method, it is proved in dimension \(3\), that one integral curve of a constant length time-like vector field with geodesic flow and with \({\mathrm{Ric}}(k,k)>0\) on a closed Lorentzian manifold is closed. This result is interesting also from the viewpoint of existence of closed geodesics on compact Lorentzian manifolds, which is open in dimension greater than or equal to \(3\).