Disconnected at infinity

A connected noncompact manifold is disconnected at infinity if removing some compact subset leaves at least two connected components that are not relatively compact in the manifold. Equivalently, the manifold has at least two ends: there are distinct ways to leave every compact subset.

The real line has two ends, while Euclidean space ${\mathbb{R}}^n$ for $n\ge 2$ has one. The circular cylinder also has two, corresponding to the two axial directions. In a proper metric space, components are not relatively compact exactly when they are unbounded, so the familiar unbounded-component formulation applies to complete connected Riemannian manifolds by the Hopf-Rinow theorem. Relative compactness gives the intrinsic definition in topology without imposing a particular metric: an open Euclidean ball is bounded in its inherited distance but still has an 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.