Hopf–Rinow theorem

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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]


Let be a connected Riemannian manifold. Then the following statements are equivalent:

  1. The closed and bounded subsets of are compact;
  2. is a complete metric space;
  3. is geodesically complete; that is, for every the exponential map expp is defined on the entire tangent space

Furthermore, any one of the above implies that given any two points 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[edit]


  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[dead link].
  3. ^ O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, p. 193, ISBN 9780080570570.