= Solution
The <Cheeger-Gromoll splitting theorem> states that a complete <connected> <Riemannian manifold> with nonnegative <Ricci curvature> that contains a <line in a Riemannian manifold> is isometric to a <Riemannian product> $N\times\mathbb R$ with the Euclidean metric on the line. The factor $N$ is complete and has nonnegative <Ricci curvature>. Under the <isometry> one can take the given line to $\{q\}\times\mathbb R$, up to translation and reversal. A line means global minimization between every pair of its points, not merely a complete <geodesic>.
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