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A strictly ordered hierarchy of eight causal properties encountered in General Relativity is reviewed for the explicit case of the gravitational plane waves. Illustrative proofs are given to the effect that the place of these space-times is precisely known in the hierarchy: they are causally continuous, but not causally simple. The other conditions of the hierarchy are also discussed separately, as are some causality conditions that belong outside the hierarchy. The investigation relies on the following tools: (1) the symplectic structure appearing in certain matrix differential equations of the Riccati type; and (2) the global properties that can be derived from the isometry group generated by the Killing vector fields of the metric. Connections to some of the techniques of singularity theory are also pointed out.

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspaceUke geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.