Hopf–Rinow theorem

Statement

1. The closed and bounded subsets of M are compact;
2. M is a complete metric space;
3. 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.

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 in M can be connected by a minimizing geodesic, and any bounded closed set in M is compact.
• The theorem does not hold in infinite dimensions: showed that two points in an infinite dimensional complete Hilbert manifold need not be connected by a geodesic.
• The theorem also does not generalize to Lorentzian manifolds: the Clifton–Pohl torus provides an example that is compact but not complete.