An $n$-dimensional Riemannian manifold $(M,g)$ is said to be geodesically complete if every geodesic $\gamma:(-\varepsilon,\varepsilon) \to M$ can be extended to a geodesic $\widetilde{\gamma}:\mathbb{R}\to M$ defined on the whole real line. There is a theorem stating that every compact manifold is geodesically complete. Can anyone provide me with a material about this theorem so that I can prove it ?.
Is the metric $g$ related to the proof?.
Best Answer
This is just a sketch of how you could prove this directly:
Let $\gamma:I\to M$ be a geodesic. Assume, for the sake of contradiction, that the maximal domain of existence is not all of $\mathbf{R}$. Say $I=(a,b)$ for some $a,b\in\mathbf{R}$. Now consider what happens to $\gamma(t)$ in the limit as $t\to b$. If $\gamma(t)$ stays in some compact set $K$ as $t\to b$, take a sequence $t_n\to b$ as $n\to\infty$. Since geodesics are constant speed, we see that $\{(t_n,\dot\gamma(t_n)\}_{n\in\mathbf{N}}$ is a compact set in the tangent bundle. So you can find some $\varepsilon > 0$ such that for every point $(p,v)$ in the compact set that there is a geodesic $\alpha:(-\varepsilon,\varepsilon)\to M$ with $\alpha(0)=p$ and $\dot\alpha(0)=v$. Now use this to extend $\gamma$ past $I=(a,b)$. Since we assume that $I$ was the maximal domain of existence for $\gamma$, we deduce that $\gamma(t)$ must leave every compact set at $t\to b$.
You now conclude by noting that if $M$ is compact, that it is impossible for any geodesic to escape every compact set. Hence, every geodesic extends for all time, and so $M$ is geodesically complete.
So in some sense the metric $g$ doesn't have much to do with the proof. The metric shows up in the geodesic differential equations, but the heart of the proof is just simple ODE theory. This result also is an immediate corollary of the Hopf-Rinow theorem.