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, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem to the context of certain types of metric spaces.

Let ${\displaystyle (M,g)}$ be a connected and smooth Riemannian manifold. Then the following statements are equivalent:

Furthermore, any one of the above implies that given any two points ${\displaystyle p,q\in M,}$ there exists a length minimizing geodesic connecting these two points (geodesics are in general critical points for the length functional, and may or may not be minima).

In the Hopf–Rinow theorem, the first characterization of completeness deals purely with the topology of the manifold and the boundedness of various sets; the second deals with the existence of minimizers to a certain problem in the calculus of variations (namely minimization of the length functional); the third deals with the nature of solutions to a certain system of ordinary differential equations.