Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifold which is a metric space
, the existence of arbitrary geodesics from
implies that
is complete with respect to
. Actually, this is slightly stronger than what H-R states: geodesic completeness at one point
implies completeness.
The first thing to notice is that is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from
exist. Moreover, it is surjective by the second Hopf-Rinow theorem.
Now fix a -Cauchy sequence
. We will show that it converges. Draw minimal geodesics
travelling at unit speed with
Then we can write
where the are unit vectors in
. Taking a subsequence, we may assume that both the
and the
(which are bounded) tend to a limit. (Recall that a Cauchy sequence converges iff a subsequence does.) By continuity of
, we find that the
tend to a limit.
From the metric-completeness criterion, we find:
Corollary 1 A compact Riemannian manifold is geodesically complete, and any two points are joined by a geodesic of minimal length.
I now want to discuss an application of some of these ideas.
Theorem 2
Letbe a topological space and
a compact Riemannian manifold. Then there exists
such that if
satisfy
![]()
for all
, then they are homotopic.
The proof of this doesn’t actually require all the machinery of the H-R theorems. The idea is that we can find a covering of
such that any two points in any
can be joined by a unique geodesic lying in
. This “normal neighborhood theorem” apparently due to Whitney is actually true without the compactness assumption and is a refinement of the usual geodesic theorem. It’s in Helgason’s book.
In the compact case, choose so small that the map
is an open imbedding for all
. (
can be chosen locally, so this is ok.) Now I claim that any two points
with
are joined by a unique geodesic in
. Uniqueness is clear from the choice of
. For existence, by H-R there is a unit-speed geodesic from
to
of length
. This is given by the image of some straight line through the origin in
that must end at a point in
, which line is thus contained in
. Thus the geodesic is contained in
.
So take this as the
in the theorem. Given
, define
by considering the unique shortest geodesic parametrized by
between
and taking the value at
. This is a homotopy; it is even smooth if
are.
In particular, if is a manifold, a continuous function
can be approximated by a smooth function
. In particular,
is homotopic to a smooth map
. This is a useful fact.
November 15, 2009 at 9:50 pm
Hi Akhil,
I have a suggestion not related to this math: since you are posting so often, it might make sense to start your own blog. It may be difficult for your fellow bloggers to post because any new posts would have to compete with yours for attention. (And also it seems to not be related to RSI anymore.)