Let M be a connected Riemannian manifold. The distance function d(x, y) (x, y ∈ M), which is by definition the inf limit of the lengths of the rectifiable arcs joining x to y, defines a metric compatible with the topology of M. By the Hopf-Rinow theorem, the following conditions are equivalent:
(i)Every geodesic can be indefinitely extended, that is, the Levi-Cività connection of M is complete (III,3.1).(ii)M is complete as a metric space.(iii)Every bounded set is relatively compact.
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