Connection by geodesics on globally hyperbolic spacetimes with a   lightlike Killing vector field
  Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauchy hypersurface. 
Rossella Bartolo, Anna Maria Candela, Jos\'e Luis Flores
math.DG