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

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: completeness, geodesic completeness, geodesics, Hopf-Rinow theorem, Riemannian manifolds

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Tags: completeness, geodesic completeness, geodesics, Hopf-Rinow theorem, Riemannian manifolds

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Ok, yesterday I covered the basic fact that given a Riemannian manifold , the geodesics on (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 is a piecewise -path between and has the smallest length among piecewise paths, then is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick very close to each other, so that is contained in a neighborhood of satisfying the conditions of yesterday’s theorem; then must be length-minimizing, so it is a geodesic. We thus see that is locally a geodesic, hence globally.

Say that is **geodesically complete** if can be defined on all of ; in other words, a geodesic can be continued to . The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)The following are equivalent:

- is geodesically complete.
- In the metric on induced by (see here), is a complete metric space

Assume the second item: let be complete in the appropriate metric. Then if for an open interval is a geodesic, consider a sequence . Then , since geodesics move at constant speed (cf. the remark after lemma 2 in the link). Thus the form a Cauchy sequence, converging to some . It is easy to check (by splicing two sequences together) that the limit does not depend on the choice of . In local coordinates we can write , when the geodesic property implies

for suitable Christoffel symbols . The first derivatives are bounded (indeed, constant), so the second derivatives are too. Thus extends to the interval with a right-handed derivative at , and moreover the right-hand derivative at exists and is uniformly continuous in a neighborhood of (by the mean value theorem). Now there is locally a geodesic at with . The function

satisfies the geodesic equation everywhere and is defined on . 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 is defined all of . Then for any , there is a geodesic from to 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 around with respect to the metric such that satisfies the conclusion of yesterday’s theorem. Take the point in (which is compact if is not too big at least) with minimized. Then

because of the definition of via lengths of curves.

There is a geodesic travelling at unit speed with , . In particular,

Let be the set of all with . The boxed statement means that , and is evidently closed. If , then we’ll be done—we’ll have a geodesic from to that minimizes length.

Since is closed, pick its largest element , and let . Choose a small neighborhood satisfying the conditions of yesterday’s theorem. Now if we pick the point in closest to , we have evidently

I claim that . First, . The path from to catenated with the geodesic from to forms a path from to of minimizing length , so it is smooth and a geodesic.

In particular, , 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.

[...] 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 [...]

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.

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.

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.

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 were a Cauchy non-convergent sequence. Wlog, there are infinitely many different terms. Then, given any , we can find a small ball around containing only finitely many of the . Since is compact, finitely many of these balls cover . So the sequence 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.)

@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.

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.

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.

Dear Sean,

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

[...] 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. [...]