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]

Statement of the theorem[edit]

Let (Mg) 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 extrema, and may or may not be minima).

Variations and generalizations[edit]

  • 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[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. 
  2. ^ 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, 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[edit]