Hopf-Rinow theorem
From Diffgeom
Statement
Let be a Riemannian manifold (viz
is a differential manifold equipped with a Riemannian metric
). Then the Hopf-Rinow theorem states the following equivalent things:
- If
is geodesically complete, then given any two points
there is a minimizing geodesic joining
and
.
- For any point
, the exponential map at
is surjective from
to
, and any length-minimizing curve from
to
for some
, occurs as the image of a straight line from the origin, under the exponential map.
Proof
- Given
, there exists
, such that the exponential map takes the unit sphere of radius
about
to the set of points on the manifold at distance
from
, with the minimal geodesics from
being exponentials of the radial paths.
The upshot of this is: given , there is an
such that the set of points at distance
from
is a compact set, and such that there is a unique minimizing geodesic from
to each of them
- Any geodesic can be extended indefinitely (and uniquely) in both directions (this follows from the assumption that our manifold is geodesically complete)
The proof beyond these two facts is purely geometric (that is, it only uses standard properties of metrics and continuous maps).