= A line from disconnection at infinity
A connected complete <Riemannian manifold> that is <disconnected at infinity> contains a <line in a Riemannian manifold>. Choose points escaping in two components outside a fixed compact set and join them by <minimizing geodesics>. Each segment crosses the compact set. Parametrize from a crossing point; both parameter endpoints tend to infinity in opposite directions. Compactness of the unit tangent vectors over the crossing set gives a subsequence converging to an entire <geodesic>. Every finite subsegment is minimizing by continuity of distance, so the limit is a line.
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