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An important, if relatively less well known aspect of the singularitytheorems in Lorentzian Geometry is to understand how their conclusions fareupon weakening or suppression of one or more of their hypotheses. Then,theorems with modified concusions may arise, showing that those conclusionswill fail only in special cases, at least some of which may be described. Theseare the so-called rigidity theorems, and have many important examples in theespecialized literature. In this paper, we prove rigidity results forgeneralized plane waves and certain globally hyperbolic spacetimes in thepresence of maximal compact surfaces. Motivated by some general propertiesappearing in these proofs, we develop the theory of lightlike sets, entitiessimilar to achronal sets, but more appropriate to deal with low-regularity nullsubmanifolds.

Lightlike sets with applications to the rigidity of null geodesic incompleteness