Hopf–Rinow theorem explained

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] Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem to the context of certain types of metric spaces.

Statement

Let

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

1. 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 \operatorname{T} _{p} M.

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

p, 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).

In the Hopf–Rinow theorem, the first characterization of completeness deals purely with the topology of the manifold and the boundedness of various sets; the second deals with the existence of minimizers to a certain problem in the calculus of variations (namely minimization of the length functional); the third deals with the nature of solutions to a certain system of ordinary differential equations.

Variations and generalizations

• The Hopf–Rinow theorem is generalized to length-metric spaces the following way:
  • If a length-metric space is complete and locally compact then any two points can be connected by a minimizing geodesic, and any bounded closed set is compact.

In fact these properties characterize completeness for locally compact length-metric spaces.

• The theorem does not hold for infinite-dimensional manifolds. The unit sphere in a separable Hilbert space can be endowed with the structure of a Hilbert manifold in such a way that antipodal points cannot be joined by a length-minimizing geodesic. It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not.
• The theorem also does not generalize to Lorentzian manifolds: the Clifton–Pohl torus provides an example (diffeomorphic to the two-dimensional torus) that is compact but not complete.

References

• • Book: 1744486. Bridson. Martin R.. Haefliger. André. Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften. 319. Springer-Verlag. Berlin. 1999. 3-540-64324-9. 10.1007/978-3-662-12494-9. Martin Bridson. André Haefliger. 0988.53001.
• Book: do Carmo, Manfredo Perdigão. Manfredo do Carmo

. Manfredo do Carmo. Riemannian geometry. Mathematics: Theory & Applications. 1992. 0-8176-3490-8. Boston, MA. Birkhäuser Boston, Inc.. Translated from the second Portuguese edition by Francis Flaherty. 0752.53001. 1138207.

• Book: Gallot. Sylvestre. Sylvestre Gallot. Hulin. Dominique. Dominique Hulin. Lafontaine. Jacques. Riemannian geometry. 2004. Third. Universitext. Springer-Verlag. 2088027. 3-540-20493-8. 10.1007/978-3-642-18855-8. 1068.53001.
• Book: Gromov. Misha. Metric Structures for Riemannian and Non-Riemannian Spaces. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. Progress in Mathematics. 152. Birkhäuser Boston, Inc.. Boston, MA. 1999. 0-8176-3898-9. 1699320. Bates. Sean Michael. With appendices by M. Katz, P. Pansu, and S. Semmes.. 10.1007/978-0-8176-4583-0. 0953.53002. Mikhael Gromov (mathematician).
• Book: Jost. Jürgen. Riemannian geometry and geometric analysis. Universitext. Jürgen Jost. Seventh edition of 1995 original. Springer, Cham. 2017. 978-3-319-61859-3. 3726907. 10.1007/978-3-319-61860-9. 1380.53001.
• Book: Shoshichi Kobayashi. Katsumi Nomizu. Kobayashi. Shoshichi. Nomizu. Foundations of Differential Geometry. Katsumi. Foundations of differential geometry. Volume I . John Wiley & Sons, Inc.. New York–London. 1963. 0152974. 0119.37502.
• Book: 1666820. Lang. Serge. Fundamentals of differential geometry. Graduate Texts in Mathematics. 191. Springer-Verlag. New York. 1999. 0-387-98593-X. 10.1007/978-1-4612-0541-8. Serge Lang. 0932.53001.
• Book: O'Neill. Barrett. Barrett O'Neill. Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics. 103. Academic Press, Inc.. New York. 1983. 0-12-526740-1. 0719023. 0531.53051. 10.1016/s0079-8169(08)x6002-7.
• Book: Petersen. Peter. Riemannian geometry. Third edition of 1998 original. Graduate Texts in Mathematics. 171. Springer, Cham. 2016. 978-3-319-26652-7. 3469435. 10.1007/978-3-319-26654-1. 1417.53001.

Notes and References

1. Hopf. H.. Rinow. W.. Ueber den Begriff der vollständigen differentialgeometrischen Fläche. Commentarii Mathematici Helvetici. 3. 1931. 1. 209–225. 10.1007/BF01601813.