Every compact manifold is geodesically complete 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?. 

I believe you are asking for the Hopf-Rinow theorem. A proof can be found in Do Carmo, "Riemannian geometry". 
