  An important, if relatively less well known aspect of the singularity
theorems in Lorentzian Geometry is to understand how their conclusions fare
upon weakening or suppression of one or more of their hypotheses. Then,
theorems with modified concusions may arise, showing that those conclusions
will fail only in special cases, at least some of which may be described. These
are the so-called rigidity theorems, and have many important examples in the
especialized literature. In this paper, we prove rigidity results for
generalized plane waves and certain globally hyperbolic spacetimes in the
presence of maximal compact surfaces. Motivated by some general properties
appearing in these proofs, we develop the theory of lightlike sets, entities
similar to achronal sets, but more appropriate to deal with low-regularity null
submanifolds.
