The Conformal Pseudodistance and Null Geodesic Incompleteness

We clarify the relationship between the null geodesic completeness of an Einstein Lorentz manifold and its conformal Kobayashi pseudodistance. We show that an Einstein manifold has at least one incomplete null geodesic if its pseudodistancfe is nontrivial. If its pseudodistance is nondegenerate, all of its null geodesics must be incomplete. Thus an Einstein manifold (M,g) has no complete null geodesic if there is a “physical metric” in the conformal class of g satisfying the null convergence and null generic conditions.