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.

The theorem is stated as follows: Let M be a Riemann manifold. Then the following statements are equivalent:

1. The closed and bounded subsets of M are compact.
2. M is a complete metric space
3. M is a complete topological space
4. M is geodesically complete; that is, for every p in M, the exponential map exp_p is defined on the entire tangent space T_pM.

Furthermore, any one of the above implies that given any two points p and q in M, there exists a geodesic connecting these two points, and that furthermore, this geodesic is of minimal length (geodesics are in general extrema, and may or may not be minima).

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-4267-2 See section 1.4.