A connected noncompact manifold is disconnected at infinity when the complement of some compact subset has at least two unbounded connected components. Equivalently, it has at least two ends.
The relevant general concept is an End; having several ends is a property of the space, not a synonym for one end.
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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