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Hopf-Rinow II and an application November 15, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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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 {(M,g)} which is a metric space {d}, the existence of arbitrary geodesics from {p} implies that {M} is complete with respect to {d}. Actually, this is slightly stronger than what H-R states: geodesic completeness at one point {p} implies completeness.

The first thing to notice is that {\exp: T_p(M) \rightarrow M} is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from {p} exist. Moreover, it is surjective by the second Hopf-Rinow theorem.

Now fix a {d}-Cauchy sequence {q_n \in M}. We will show that it converges. Draw minimal geodesics {\gamma_n} travelling at unit speed with

\displaystyle \gamma_n(0)=p, \quad \gamma_n( d(p,q_n)) = q_n. 

Then we can write

\displaystyle q_n = \gamma_n(d(p,q_n)) = \exp_p( d(p,q_n) v_n)  

where the {v_n} are unit vectors in {T_p(M)}. Taking a subsequence, we may assume that both the {v_n} and the {d(p,q_n)} (which are bounded) tend to a limit. (Recall that a Cauchy sequence converges iff a subsequence does.) By continuity of {\exp_p}, we find that the {q_n} 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

Let {X} be a topological space and {M} a compact Riemannian manifold. Then there exists {\epsilon>0} such that if {f,h: X \rightarrow M} satisfy\displaystyle d(f(x),g(x))<\epsilon  

for all {x \in X}, 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 {U_i} of {M} such that any two points in any {U_i} can be joined by a unique geodesic lying in {U_i}. 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 {r>0} so small that the map {\exp_p: B_r(0) \subset T_p(M) \rightarrow M} is an open imbedding for all {p}. ({r} can be chosen locally, so this is ok.) Now I claim that any two points {p,q \in M} with {d(p,q)<r} are joined by a unique geodesic in {D_r(p) \cap D_r(q) \subset M}. Uniqueness is clear from the choice of {r}. For existence, by H-R there is a unit-speed geodesic from {p} to {q} of length {d(p,q)}. This is given by the image of some straight line through the origin in {T_p(M)} that must end at a point in {B_r(0)}, which line is thus contained in {B_r(0)}. Thus the geodesic is contained in {D_r(p)}.

So take this {r} as the {\epsilon} in the theorem. Given {f,h}, define {H(x,t)} by considering the unique shortest geodesic parametrized by {[0,1]} between {f(x),h(x)} and taking the value at {t}. This is a homotopy; it is even smooth if {f,h} are.

In particular, if {X} is a manifold, a continuous function {f: X \rightarrow M} can be approximated by a smooth function {h: X \rightarrow M}. In particular, {f} is homotopic to a smooth map {h: X \rightarrow M}. This is a useful fact.

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Comments»

1. Steven Sam - November 15, 2009

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.)

Akhil Mathew - November 15, 2009

Heh.

You’re of course right that a MaBloWriMo series on differential geometry has essentially nothing to do with the purpose of this blog (“Mathematical research and problem-solving,” RSI, USAMO, etc.), and I have already noticed that this hasn’t really been the focus for the past few months.

That said, I have been fairly busy with other activities and it will probably take me a few days to set everything up (given that I will still be aiming for one post a day). I’ll remain a contributor here, just for a different flavor of posts.


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