  The aim of this paper is to study foliations that remain invariant by
parallel transports along the integral curves of vector fields of another
foliations. According to this idea, we define a new concept of stability
between foliations. A particular case of stability (called regular stability)
is studied, giving a useful characterization in terms of the Riemann curvature
tensor. This characterization allows us to prove that there are no regularly
self-stable foliations of dimension greater than 1 in Schwarzschild and
Robertson-Walker space-times. Finally, we study the existence of regularly
self-stable foliations in other space-times, like $pp$-wave space-times.
