Geometric description of lightlike foliations by an observer in general relativity

We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension $n$, from a purely geometric point of view. Given an observer and a lightlike distribution $\Omega$ of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field $U$ representing the observer and a spacelike vector field $S$ representing the relative direction of propagation of $\Omega$ for this observer. A new distribution $\Omega_U^-$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\Omega$ and $\Omega _U^-$ are integrable, the pair \Omega ,\Omega_U^- $represents the wave fronts of a stationary wave for the observer$U$. However, we show in an example that the integrability of$\Omega $does not imply the integrability of$\Omega_U^-$.