= Ricci-flat obstruction for a closed three-manifold times a line
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If a connected <closed manifold> $M^3$ has a noncontractible <universal cover>, then $M\times\mathbb R$ has no complete metric making it a <Ricci-flat Riemannian manifold>. The product has two ends and hence contains a <line in a Riemannian manifold> for any complete metric. The <Cheeger-Gromoll splitting theorem> would split it isometrically as $N^3\times\mathbb R$. Its three-dimensional factor is Ricci-flat and therefore flat by <three-dimensional curvature from the Ricci tensor>. The complete simply connected cover of the product would be Euclidean four-space. But that cover is $\widetilde M\times\mathbb R$, which retracts onto $\widetilde M$, contradicting noncontractibility.
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