Geodesic connectedness of affine manifolds

Lorentzian distance on the Lobachevsky plane *

“…Remark 10. A Lorentzian metric on a manifold M is called geodesically connected [17,18] if any two points in M can be connected by a future directed nonspacelike geodesic. For any left-invariant Lorentzian metric on Aff + (R) we have J + (Id) = Aff + (R), so such a metric is not geodesically connected.…”

Causal simplicity and (maximal) null pseudoconvexity

“…Geodesic nonimprisonment and pseudoconvexity of all geodesics together with the absence of conjugate points implies geodesic connectedness and serve as conditions for a pseudo-Riemannian version of the Hadamard-Cartan theorem [7], [5, chapter 11]. For a recent generalization see [11].…”