Cheeger-Gromoll splitting theorem

This is a Riemannian splitting theorem whose hypotheses include a globally minimizing geodesic line.

A complete connected Riemannian manifold with nonnegative Ricci curvature that contains a line splits isometrically as $N\times \mathbb{R}$.

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 $\tilde{M}\times \mathbb{R}$, which retracts onto $\tilde{M}$, contradicting noncontractibility.