Hopf-Rinow theorem | Climbing Mount Bourbaki

Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifold ${(M,g)}$ which is a metric space ${d}$ , the existence of arbitrary geodesics from ${p}$ implies that ${M}$ is complete with respect to ${d}$ . Actually, this is slightly stronger than what H-R states: geodesic completeness at one point ${p}$ implies completeness.

The first thing to notice is that ${\exp: T_p(M) \rightarrow M}$ is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from ${p}$ exist. Moreover, it is surjective by the second Hopf-Rinow theorem.

Now fix a ${d}$ -Cauchy sequence ${q_n \in M}$ . We will show that it converges. Draw minimal geodesics ${\gamma_n}$ travelling at unit speed with

$\displaystyle \gamma_n(0)=p, \quad \gamma_n( d(p,q_n)) = q_n.$ (more…)

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 (more…)

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Climbing Mount Bourbaki

Hopf-Rinow II and an application

The Hopf-Rinow theorems and geodesic completeness

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