Hopf–Rinow theorem
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.
Statement
Let be a connected and smooth Riemannian manifold. Then the following statements are equivalent:- The closed and bounded subsets of are compact;
- is a complete metric space;
- is geodesically complete; that is, for every the exponential map expp is defined on the entire tangent space
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 ; the third deals with the nature of solutions to a certain system of ordinary differential equations.
Variations and generalizations
- The Hopf–Rinow theorem is generalized to length-metric spaces the following way:
- * If a length-metric space is complete and locally compact then any two points can be connected by a minimizing geodesic, and any bounded closed set is compact.
- The theorem does not hold for infinite-dimensional manifolds. The unit sphere in a separable Hilbert space can be endowed with the structure of a Hilbert manifold in such a way that antipodal points cannot be joined by a length-minimizing geodesic. It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not.
- The theorem also does not generalize to Lorentzian manifolds: the Clifton–Pohl torus provides an example that is compact but not complete.