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

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 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:
    If a length-metric space (M, d) 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: (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. hdl:10338.dmlcz/101427.
  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.
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References
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