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.
