Suvarna Garge (Editor)

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.

Contents

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

    • References

    Hopf–Rinow theorem Wikipedia
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