High Quality Content by WIKIPEDIA articles! In mathematics, the Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow (1907–1979).Furthermore, any one of the above implies that given any two points p and q in M, there exists a length minimizing geodesic connecting these two points (geodesics are in general extrema, and may or may not be minima). This does not hold in infinite dimensions: (Atkin 1975) showed that two points in an infinite dimensional complete Riemannian manifold need not be connected by any geodesic.

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