Geodesics in Lorentzian Manifolds

A geodesic on a Riemannian manifold is, locally, a length minimizing curve. For example, a geodesic in the Euclidean plane is a straight line and on the sphere, all geodesics are great circles. We notice that it is positive definite(Riemannian). Moreover, A connected Riemannian manifold is geodesically complete if and only if it is complete as a metric space. Manifolds whose metric is not positive definite (pseudo-Riemannian). Since the distance function is no longer positive definite and geodesics here can be viewed as a distance between events.They are no longer distance minimizing instead, some are distance maximizing or zero. A manifold is geodesically complete if every geodesic extends for infinite time.There are three types of geodesic completeness on pseudo-Riemannian manifolds:Timelike geodesically complete if every timelike geodesic extends for infinite time. Spacelike geodesically complete if every timelike geodesic extends for infinite time. Lightlike geodesically complete if every timelike geodesic extends for infinite time. It is our goal to show that these notions are inequivalent.