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The Hopf-Rinow theorems and geodesic completeness November 14, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Ok, yesterday I covered the basic fact that given a Riemannian manifold {(M,g)}, the geodesics on {M} (with respect to the Levi-Civita connection) locally minimize length. Today I will talk about the phenomenon of “geodesic completeness.”

Henceforth, all manifolds are assumed connected.

The first basic remark to make is the following. If {c: I \rightarrow M} is a piecewise {C^1}-path between {p,q} and has the smallest length among piecewise {C^1} paths, then {c} is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick {a,b \in I} very close to each other, so that {c([a,b])} is contained in a neighborhood of {c\left( \frac{a+b}{2}\right)} satisfying the conditions of yesterday’s theorem; then {c|_{[a,b]}} must be length-minimizing, so it is a geodesic. We thus see that {c} is locally a geodesic, hence globally.

Say that {M} is geodesically complete if {\exp} can be defined on all of {TM}; in other words, a geodesic {\gamma} can be continued to {(-\infty,\infty)}. The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)

The following are equivalent:

  • {M} is geodesically complete.
  • In the metric {d} on {M} induced by {g} (see here), {M} is a complete metric space

 

Assume the second item: let {M} be complete in the appropriate metric. Then if {\gamma: I \rightarrow M} for {I} an open interval {(a,b)} is a geodesic, consider a sequence {b_n \rightarrow b}. Then {d(\gamma(b_n),\gamma(b_{m})) \leq l( \gamma|_{[b_n,b_m]}) = O(|b_n-b_m|)}, since geodesics move at constant speed (cf. the remark after lemma 2 in the link). Thus the {\gamma(b_n)} form a Cauchy sequence, converging to some {p \in M}. It is easy to check (by splicing two sequences together) that the limit does not depend on the choice of {\{b_n\}}. In local coordinates we can write {\gamma=(\gamma_1, \dots, \gamma_n)}, when the geodesic property implies

\displaystyle \dot{\dot{\gamma_i}} = -\sum_{j,k} \Gamma^i_{jk} \dot{\gamma_j}\dot{\gamma_k}

for suitable Christoffel symbols {\Gamma^i_{jk}}. The first derivatives {\dot{\gamma_j}} are bounded (indeed, constant), so the second derivatives are too. Thus {\gamma} extends to the interval {(a,b]} with a right-handed derivative {\dot{\gamma}} at {b}, and moreover the right-hand derivative at {b} exists and is uniformly continuous in a neighborhood of {b} (by the mean value theorem). Now there is locally a geodesic {\gamma_1:(b-\epsilon,b+\epsilon)} at {p} with {\dot{\gamma_1}(b) = \dot{\gamma}(b)}. The function

\displaystyle \gamma_2(t):= \begin{cases} \gamma(t) & \text{if } t \leq b \\ \gamma_1(t) & \text{otherwise} \end{cases}  

satisfies the geodesic equation everywhere and is defined on {(a,b+\epsilon)}. So we can extend these geodesics to the right, and similarly it is done to the left.

To go the other way, we first prove another Hopf-Rinow theorem, interesting in its own right:

Theorem 2 (Hopf-Rinow) Suppose {\exp_p} is defined all of {T_p(M)}. Then for any {q \in M}, there is a geodesic {\gamma} from {p} to {q} that minimizes length.

The proof is really a nice bit of geometry. This is a global result, unlike yesterday’s theorem.

Consider a small sphere {S_r(p)} around {p} with respect to the metric {d} such that {D_{2r}(p)} satisfies the conclusion of yesterday’s theorem. Take the point {p'} in {S_r(p)} (which is compact if {r} is not too big at least) with {d(p',q)} minimized. Then

\displaystyle d(p,q) = d(p',q) + r  

because of the definition of {d} via lengths of curves.

HR1

There is a geodesic {\gamma} travelling at unit speed with {\gamma(0)=p}, {\gamma(r)=p'}. In particular,

\displaystyle \boxed{ d( \gamma(r), q) = d(p,q) - r.} 

Let {S} be the set of all {s} with {d( \gamma(s), q) = d(p,q) - s}. The boxed statement means that {r \in S}, and {S} is evidently closed. If {d(p,q) \in S}, then we’ll be done—we’ll have a geodesic from {p} to {q} that minimizes length.

Since {S \cap [0,d(p,q)]} is closed, pick its largest element {s \in S \cap [0,d(p,q))}, and let {u = \gamma(s)}. Choose a small neighborhood {D_{2\delta}(u)} satisfying the conditions of yesterday’s theorem. Now if we pick the point {u'} in {S_{\delta}(u)} closest to {q}, we have evidently

\displaystyle d(u',q) = d(u,q) - \delta = d(p,q) - s - \delta.

 HR2

I claim that {u' = \gamma(s+\delta)}. First, {d(p,u') \geq d(q,p) - d(q,u') = d(p,q) - s + \delta}. The path {\gamma} from {p} to {u} catenated with the geodesic from {u} to {u'} forms a path from {p} to {u'} of minimizing length {d(p,q) + \delta = d(p,q) - s + \delta}, so it is smooth and a geodesic.

In particular, {s+\delta \in S}, so we get a contradiction and hence the second theorem.

Since this post has already reached a certain length, I’ll defer the proof of the second implication in the first Hopf-Rinow theorem for tomorrow.  Also, I should add that I’ve followed Milnor’s Morse Theory, chapter 2, in the proof of the second H-R theorem.

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Comments»

1. Hopf-Rinow II and an application « Delta Epsilons - November 15, 2009

[...] manifolds trackback Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifodl which is a metric space , the existence of [...]

2. Anirbit - January 22, 2011

Can you emphasize on the role of the compactness issue in the Hopf-Rinow theorem.

If I am not wrong then the following are also true,

* That given a non-compact manifold one can always put a complete Riemannian metric. (intuitively making the metric hyperbolic near the missing points)

* That given a compact manifold you can never put an incomplete Riemannian metric on it.

It would be nice if you can elaborate on these and their relationship to the Hopf-Rinow theorem.

Akhil Mathew - January 22, 2011

Dear Anirbit,

I suspect your first assertion is true. It would appear that the way to prove it would be to start at a given point and with a given metric, then start vastly increasing the metric as one goes outward, so that geodesics (which travel at unit speed) cannot “escape to infinity” in finite time.

Your second assertion is definitely true. A compact manifold is always complete (i.e. Cauchy complete, with *any* metric inducing the same topology, e.g. the associated metric to the Riemannian 2-tensor), so that by the Hopf-Rinow theorem geodesics extend as long as you want.

3. Anirbit - January 22, 2011

Dear Akhil,

Thanks for your reply.

Is “inducing the same topology” a required caveat if one is working on Riemannian manifolds?

I think given a manifold any Riemannian metric on it will induce the same topology and the same as the intrinsic manifold topology.

Personally I alway think of Hopf-Rinow theorem as saying that you cannot put incomplete metrics on compact Riemannian manifolds.

(it also says that completeness in the induced metric is the same as geodesic completeness)

I guess for general topological spaces the second assertion should fail. I think there should be examples of compact spaces with incomplete metrics. It would be nice to see examples of this kind.

Akhil Mathew - January 22, 2011

Dear Anirbit,

The metric associated to any Riemannian metric induces the given topology on the underlying manifold. (If you think in euclidean space, this follows because locally, the norm induced by a Riemannian metric will differ by some scalar from the usual norm.) So your second assertion is correct, and my statement earlier was redundant (it was made more for emphasis).

The Hopf-Rinow theorem is still applicable to non-constant manifolds, though, and sometimes one wishes to consider them! (See for instance http://amathew.wordpress.com/2009/11/28/the-ricci-tensor-and-manifolds-of-positive-curvature/)

For any compact metric topological space, the metric is automatically complete. (Proof: suppose \{x_n\} were a Cauchy non-convergent sequence. Wlog, there are infinitely many different terms. Then, given any x \in X, we can find a small ball around x containing only finitely many of the \{x_n\}. Since X is compact, finitely many of these balls cover X. So the sequence \{x_n\} consisted of a finite set, contradiction.) The general result is that in a topological space, one has compactness iff every filter has a limit point. (Having limits of sequences may not be enough to ensure compactness.)

4. Anirbit - January 23, 2011

@Akhil

Isn’t the statement that you are making that for a compact metric space every metric is complete true only if you consider metrics whose induced topology coincides with the one you started with?

I was only saying that you can always put an incomplete metric on your compact space just that its induced topology will be different.

Akhil Mathew - January 23, 2011

Oh! Yes, of course. You can always put (say) the discrete topology on an infinite compact space, and the new metric will induce a different topology with respect to which the new space *won’t* be compact.

5. Sean - October 31, 2011

Been a while since this was posted, but I stumbled on it and was wondering if you knew anything about geodesics for metrics with low regularity – in particular for metrics that are only Lipschitz continuous. These would have to be defined as local length minimizing curves (since the relevant ODE would only have L^infty coefficients, and therefore would not be well posed). I was wondering if there is any equivalent to the Hopf-Rinow theorem for that case.

Akhil Mathew - November 1, 2011

Dear Sean,

That sounds like an interesting question. Unfortunately I have no idea how to answer it; perhaps you should try MathOverflow!

6. 1. Hopf–Rinow theorem | Daily Meditations - February 2, 2012

[...] C.J.Aitken proved it to be false in infinite dimension (first page of the PDF is here) and  one can read the 15-page paper by Ivar Ekeland on the generalization.  Shlomo Sternberg has some an invaluable link to some slides on this topic. Gliklikh also has a paper about generalizations of the theorem in geodesics. ArXiv has a paper of the theorem in Sato-Grassmannian. Last but not the least, there is Akhil Matthew’s blog on this topic. [...]


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