Light-like parallel vector fields and Einstein’s equations of gravity

We prove that, contrary to the situation with time-like and space-like parallel vector fields, there are real gravitational fields satisfying Einstein’s equations of gravity and admitting non-trivial light-like parallel vector fields; we solve completely the field equations under this geometric constraint and obtain through Bianchi identities an interesting property for the Riemann curvature tensor corresponding with a real gravitational field admitting a non-trivial parallel vector field. The solutions we obtained turn out to be generically wave-like and then we prove by a geometric method that this class of solutions is quite different from the class of solutions corresponding with spherical waves.