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.