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…)