Hopf-Rinow Theorem

Let  be a Riemannian manifold, and let the topological metric on  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.  is geodesically complete, i.e., all geodesics are defined for all time.

2.  is geodesically complete at some point , i.e., all geodesics through  are defined for all time.

3.  satisfies the Heine-Borel property, i.e., every closed bounded set is compact.

4.  is metrically complete.

REFERENCES:

Petersen, P. Riemannian Geometry. New York: Springer Verlag, p. 125, 1998.