= Hopf-Rinow theorem
{c}
{wiki=Hopf–Rinow_theorem}
For a connected finite-dimensional <Riemannian manifold>, completeness of the <Riemannian distance>, <geodesic completeness>, and <compactness> of every closed bounded subset are equivalent. These conditions imply that any two points are joined by a <minimizing geodesic>. Pairwise existence of <minimizing geodesics> alone does not imply completeness: an open <Euclidean ball> is incomplete, yet the <line segment> between any two of its points stays in the ball and minimizes length.
Back to article page
