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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.

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

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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.

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  • $\begingroup$ Oh wow can't believe I didn't think of that. $\endgroup$
    – ಠ_ಠ
    Mar 26, 2019 at 7:11

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