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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