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 $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ for this observer. A new distribution ${\mathrm{\Omega }}_U^{-}$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\mathrm{\Omega }$ and $\Omega _U^-$areintegrable,thepair\mathrm{\Omega },{\mathrm{\Omega }}_U^{-}$ represents the wave fronts of a stationary wave for the observer $U$. However, we show in an example that the integrability of $\mathrm{\Omega }$ does not imply the integrability of ${\mathrm{\Omega }}_U^{-}$.