# Universal cover of a not-necessarily-complete hyperbolic manifold

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.

• You answered your own question by puncturing a complete hyperbolic $n$-manifold (if $n>2$). Mar 26, 2019 at 2:46
• @MoisheKohan Could you please elaborate? I'm not seeing how that answers the question.
– ಠ_ಠ
Mar 26, 2019 at 2:48
• Ah I see now; don't know how I missed that.
– ಠ_ಠ
Mar 26, 2019 at 7:11

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.