I have to prove that every homogeneous manifold is complete.
Let $p,q$ points on a Riemannian homogeneous manifold. I want to prove that there exist a minimizing geodesic join $p,q$ or equivalently, that every geodesic can be extended forever, that is the same that the $\exp_p$ is defined to all $T_pM$ for at least one point $p$.
My idea is the following: We know that there is an open ball in $T_pM$ with some radius $\epsilon > 0$ such that $\exp_p$ is a diffeo in some open set on $M$. In particular, the image of the diffeo has the property that every point can be join to $p$ by a minimizing geodesic with length less than $\epsilon.$
Once $M$ is homogeneous, then every point admits a ball with the same radius and same property. This can seen noting that given two arbitrary point there is an isometry that "connects" both. Then I can transfer the geodesics from a point to another.
Ok, then:
For every point I can produce a ball with radius $\epsilon$ with the property that for every point on the ball there exist a minimizing geodesic join two points.
How can I conclude then that I can extend every geodesic? I know that for every point I can produce geodesics with the same length, how to finish?
