Lorentzian compact manifolds: isometries and geodesics

In this work we investigate families of compact Lorentzian manifolds in dimension four. We show that every lightlike geodesic on such spaces is periodic, while there are closed and non-closed spacelike and timelike geodesics. Their isometry groups are computed. We also show that there is a non trivial action by isometries of $\Heis_3(\RR)$ on the nilmanifold $S^1\times (\Gamma_k \bsh \Heis_3(\RR))$ for $\Gamma_k$ a lattice of $\Heis_3(\RR)$.