On the Geometry of pp-Wave Type Spacetimes

Global geometric properties of product manifolds M = M × &R;2, endowed with a metric type <·,·> = <·,·> R + 2dudv + H(x,u)du2 (where <·,·> R is a Riemannian metric on M and H : M×&R; → &R; a function), which generalize classical plane waves, are revisited. Our study covers causality (causal ladder, non-existence of horizons), geodesic completeness, geodesic connectedness and existence of conjugate points. Appropriate mathematical tools for each problem are emphasized and the necessity to improve several Riemannian (positive de.nite) results is claimed.