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:

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_pM.

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

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.
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.). Universitext. Springer Science & Business Media. doi:10.1007/978-3-642-21298-7. ISBN 978-3-642-21298-7. See section 1.7.
Voitsekhovskii, M. I. (2001) [1994], "Hopf-Rinow theorem", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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[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.