Hopf-Rinow II and an application November 15, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: geodesic completeness, geodesics, homotopy, Hopf-Rinow theorem, Riemannian manifolds
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 2Let be 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.