Hopf–Rinow theorem

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.^[1]

Statement of the theorem

Let (M, g) be a connected Riemannian 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 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 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:

  If a length-metric space (M, d) is complete and locally compact then any two points in M can be connected by minimizing geodesic, and any bounded closed set in M is compact.

• 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.^[2]
• The theorem also does not generalize to Lorentzian manifolds: the Clifton–Pohl torus provides an example that is compact but not complete.^[3]

Notes

1. Hopf, H.; Rinow, W. (1931). "Ueber den Begriff der vollständigen differentialgeometrischen Fläche". Commentarii Mathematici Helvetici. 3 (1): 209–225. doi:10.1007/BF01601813.
2. 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, doi:10.1112/blms/7.3.261, MR 0400283.
3. O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, p. 193, ISBN 9780080570570.

References

• Jürgen Jost (28 July 2011). Riemannian Geometry and Geometric Analysis (6th Ed.). Springer Science & Business Media. ISBN 978-3-642-21298-7. See section 1.7.
• Voitsekhovskii, M. I. (2001), "H/h048010", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4