On homogeneous plane waves
Résumé
Plane waves are a special class of Lorentzian spaces and of a great importance in physics and in general relativity. Here we consider $1$-connected non-flat homogeneous plane waves, and we essentially establish a correspondence between these spaces and certain $1$-parameter groups of automorphisms of Heisenberg. It is well known that plane waves admit an infinitesimal Heisenberg action. We show here that this action integrates to a global one under the homogeneity assumption, and as a consequence, we obtain the form of the identity component of the isometry group of a $1$-connected non-flat homogeneous plane wave. Using the Heisenberg global action, we show that a $1$-connected homogeneous plane wave has global Brinkmann coordinates.