The Hopf-Rinow theorem applies to a connected Riemannian manifold 
without boundary. Equip 
with the distance defined as
the infimum of the arc lengths
of piecewise smooth curves joining two points.
Then the following are equivalent (Burtscher 2025, p. 76):
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.
The absence of boundary matters. The closed interval ![[0,1]](/images/equations/Hopf-RinowTheorem/Inline9.svg)
with its usual distance is a complete
metric space, but an interior unit-speed geodesic
reaches an endpoint in finite time and cannot extend
within the interval for all time.
