# 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 of the theorem

Let (*M*, *g*) be a connected Riemannian manifold. Then the following statements are equivalent:

- The closed and bounded subsets of
*M*are compact; -
*M*is a complete metric space; -
*M*is geodesically complete; that is, for every*p*in*M*, the exponential map exp_{p}is defined on the entire tangent space T_{p}*M*.

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 minimizing geodesic, and any bounded closed set in*M*is compact.

- If a length-metric space (
- 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

- ↑ Hopf, H.; Rinow, W. (1931). "Ueber den Begriff der vollständigen differentialgeometrischen Fläche".
*Commentarii Mathematici Helvetici*.**3**(1): 209–225. doi:10.1007/BF01601813. - ↑ 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. - ↑ O'Neill, Barrett (1983),
*Semi-Riemannian Geometry With Applications to Relativity*, Pure and Applied Mathematics,**103**, Academic Press, p. 193, ISBN 9780080570570.

## References

- Jürgen Jost (28 July 2011).
*Riemannian Geometry and Geometric Analysis (6th Ed.)*. Springer Science & Business Media. ISBN 978-3-642-21298-7. See section 1.7*.* - Voitsekhovskii, M. I. (2001), "H/h048010", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4

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