If $M=\mathbb{H}^n/\Gamma$ is a hyperbolic manifold ($\Gamma$ being a group of isometries of $\mathbb{H}^n$) can I conclude that it is complete?
I know that if the covering map was a local isometry then $M$ would be complete. But since $M$ is a hyperbolic manifold, it is locally isometric to $\mathbb{H}^n$. However, I am not able to conclude that the covering map is a local isometry, and I have not found any counterexample.