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.

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

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

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.