The Cartan-Hadamard theorem (as typically stated) tells us that the universal cover of a geodesically complete and connected Riemannian manifold $M$ with non-positive sectional curvature is diffeomorphic to $\mathbb{R}^n$.
If $M$ is a hyperbolic manifold (constant negative sectional curvature) which is incomplete, can we still conclude that the universal cover will be diffeomorphic to $\mathbb{R}^n$? If so, is there a reference for this fact?
For example, we could take a complete hyperbolic manifold and puncture it by removing a point to obtain an incomplete one, or we could cut the complete hyperbolic manifold in two along a separating hypersurface if one exists.
Take the hyperbolic plane $\mathbb H^n$ and remove one point. If $n> 2$ it is simply connected and thus is the universal cover of itself.