Statement of the theorem
Let (M, g) be a connected Riemannian manifold. Then the following statements are equivalent:
- The closed and bounded subsets of M are compact;
- M is a complete metric space;
- M is geodesically complete; that is, for every p in M, the exponential map expp is defined on the entire tangent space TpM.
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 critical points for the length functional, and may or may not be minima).
Variations and generalizations
- The Hopf–Rinow theorem is generalized to length-metric spaces the following way:
- The theorem does not hold in infinite dimensions: (Atkin 1975) 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.
- Hopf, H.; Rinow, W. (1931). "Ueber den Begriff der vollständigen differentialgeometrischen Fläche". Commentarii Mathematici Helvetici. 3 (1): 209–225. doi:10.1007/BF01601813.
- Atkin, C. J. (1975), "The Hopf–Rinow theorem is false in infinite dimensions" (PDF), The Bulletin of the London Mathematical Society, 7 (3): 261–266, MR 0400283, doi:10.1112/blms/7.3.261.
- O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, p. 193, ISBN 9780080570570.