Hopf–Rinow theorem

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 (1907–1979).

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 extrema, 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.

References

• Atkin, C. J. (1975), "The Hopf–Rinow theorem is false in infinite dimensions", The Bulletin of the London Mathematical Society 7 (3): 261–266, doi:10.1112/blms/7.3.261, MR0400283, http://blms.oxfordjournals.org/cgi/reprint/7/3/261.pdf 
• Hopf, H., Rinow, W., Über den Begriff der vollständigen differentialgeometrischen Fläche, Comment. Math. Helv. 3 (1931), 209–225.
• Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2See section 1.4.
• Voitsekhovskii, M.I. (2001), "Hopf–Rinow theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/H/h048010.htm