Let M be a Riemannian manifold, and let the topological metric on M be defined by letting the distance between two points be the infimum of the lengths of curves joining the two points. The Hopf-Rinow theorem then states that the following are equivalent: 1. M is geodesically complete, i.e., all geodesics are defined for all time. 2. M is geodesically complete at some point p, i.e., all geodesics through p are defined for all time. 3. M satisfies the Heine-Borel property, i.e., every closed bounded set is compact. 4. M is metrically complete.
