When is the universal cover of a Riemannian manifold complete?

Question

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I want to know under what conditions the universal cover $\tilde{M}$ is complete. The reason for this questions is that I want to know under what conditions on $M$ the Hopf-Rinow theorem can be applied to the universal cover.

On Wolfram (http://mathworld.wolfram.com/CompleteRiemannianMetric.html) it says that if $M$ is compact, its universal cover is complete. Would someone be able to give a proof of this?

And what deductions can we make if $M$ is complete (and possibly fulfills some other conditions)? (I'm not really looking for curvature conditions like corollaries of the Bonnet-Myers theorem).

Thanks in advance for any help!

Answer 1

It's actually true that $M$ is complete if and only if its universal cover $\widetilde{M}$ is complete. Let $p: \widetilde{M} \to M$ be the universal covering map, $q \in M$ and $\tilde{q} \in p^{-1}(q)$. As has already been stated, by Hopf-Rinow, all we need to if we want to conclude that $\widetilde{M}$ implies $M$ is complete is prove the corresponding statement for the exponential maps based at $q$ and $\tilde{q}$.

Now if $\widetilde{M}$ is complete and $\widetilde{E}: T_{\widetilde{q}}\widetilde{M} \to \widetilde{M}$ is its exponential map based at $\widetilde{q}$, define a map

$$ E = p \circ \widetilde{E} \circ (dp)^{-1}: T_qM \to M$$

You can show that $p$ sends geodesics to geodesics by showing their images are locally length minimizing. Since $(dp)^{-1}$ is linear it sends radial lines to radial lines, and you can use this to show that $E$ is exactly the exponential map for $M$ based at $p$. Then from the above it follows immediately that $E$ is defined on the whole tangent space.

The other answer shows the reverse implication.

Answer 2

If $M$ is compact,Hopf Rinow implies that $M$ is complete, let $\hat M$ be the universal cover of $M$, and $p:\hat M\rightarrow M$.Lift the metric defined on $M$ with $p$ and $p:\hat M\rightarrow M$ preserves the metric.Suppose $\hat M$ is incomplete. Let $c:I=(a,b)\rightarrow \hat M$ an incomplete geodesic maximal that you can't extend. $p(c)$ can be extended to $c':(a,b+\epsilon)\rightarrow M$ since $M$ is complete, and you can lift $c'$ to a geodesic $(a,b+\epsilon)\rightarrow \hat M$ since $p$ is a covering map. Contradiction.