On the existence of geodesics on globally hyperbolic spacetimes with a lightlike Killing vector field
By Rosella Bartolo (Politecnico di Bari)
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.
 R. Bartolo, A.M. Candela, J.L. Flores: Connection by geodesics on globally hyperbolic spacetimes with a lightlike Killing vector field. arXiv:1405.0804[math.DG], (2014).
 A.M. Candela, J.L. Flores, M. Sánchez: Global hyperboliticity and Palais–Smale condition for action functionals in stationary spacetimes. Adv. Math. 218, 515-536 (2008).
 F. Giannoni, P. Piccione: An intrinsic approach to the geodesical connectedness of stationary Lorentzian manifolds. Comm. Anal. Geom. 1, 157-197 (1999).