  Global geometric properties of product manifolds ${\cal M}= M \times \R^2$,
endowed with a metric type $<\cdot, \cdot > = < \cdot, \cdot >_R + 2 dudv +
H(x,u) du^2$ (where $<\cdot, \cdot >_R$ is a Riemannian metric on $M$ and $H:M
\times \R \to \R$ a function), which generalize classical plane waves, are
revisited. Our study covers causality (causal ladder, inexistence of horizons),
geodesic completeness, geodesic connectedness and existence of conjugate
points. Appropiate mathematical tools for each problem are emphasized and the
necessity to improve several Riemannian (positive definite) results is claimed.
