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] Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem to the context of certain types of metric spaces.
Statement
Let be a connected and smooth Riemannian manifold. Then the following statements are equivalent:[2]
- The closed and bounded subsets of are compact;
- is a complete metric space;
- is geodesically complete; that is, for every the exponential map expp is defined on the entire tangent space
Furthermore, any one of the above implies that given any two points 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:[3]
- 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.[4]
- 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.[5] It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not.[6]
- 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.[7]
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. S2CID 117624215.
- do Carmo 1992, Chapter 7; Gallot, Hulin & Lafontaine 2004, Section 2.C.5; Jost 2017, Section 1.7; Kobayashi & Nomizu 1963, Section IV.4; Lang 1999, Section VIII.6; O'Neill 1983, Theorem 5.21 and Proposition 5.22; Petersen 2016, Section 5.7.1.
- Bridson & Haefliger 1999, Proposition I.3.7; Gromov 1999, Section 1.B.
- Burago, Burago & Ivanov 2001, Section 2.5.3.
- Lang 1999, pp. 226–227.
- 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
- Gallot, Hulin & Lafontaine 2004, Section 2.D.4; O'Neill 1983, p. 193.
References
- Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001). A course in metric geometry. Graduate Studies in Mathematics. Vol. 33. Providence, RI: American Mathematical Society. doi:10.1090/gsm/033. ISBN 0-8218-2129-6. MR 1835418. Zbl 0981.51016. (Erratum: )
- Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften. Vol. 319. Berlin: Springer-Verlag. doi:10.1007/978-3-662-12494-9. ISBN 3-540-64324-9. MR 1744486. Zbl 0988.53001.
- do Carmo, Manfredo Perdigão (1992). Riemannian geometry. Mathematics: Theory & Applications. Translated from the second Portuguese edition by Francis Flaherty. Boston, MA: Birkhäuser Boston, Inc. ISBN 0-8176-3490-8. MR 1138207. Zbl 0752.53001.
- Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004). Riemannian geometry. Universitext (Third ed.). Springer-Verlag. doi:10.1007/978-3-642-18855-8. ISBN 3-540-20493-8. MR 2088027. Zbl 1068.53001.
- Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Vol. 152. Translated by Bates, Sean Michael. With appendices by M. Katz, P. Pansu, and S. Semmes. (Based on the 1981 French original ed.). Boston, MA: Birkhäuser Boston, Inc. doi:10.1007/978-0-8176-4583-0. ISBN 0-8176-3898-9. MR 1699320. Zbl 0953.53002.
- Jost, Jürgen (2017). Riemannian geometry and geometric analysis. Universitext (Seventh edition of 1995 original ed.). Springer, Cham. doi:10.1007/978-3-319-61860-9. ISBN 978-3-319-61859-3. MR 3726907. Zbl 1380.53001.
- Kobayashi, Shoshichi; Nomizu, Katsumi (1963). Foundations of differential geometry. Volume I. New York–London: John Wiley & Sons, Inc. MR 0152974. Zbl 0119.37502.
- Lang, Serge (1999). Fundamentals of differential geometry. Graduate Texts in Mathematics. Vol. 191. New York: Springer-Verlag. doi:10.1007/978-1-4612-0541-8. ISBN 0-387-98593-X. MR 1666820. Zbl 0932.53001.
- O'Neill, Barrett (1983). Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics. Vol. 103. New York: Academic Press, Inc. doi:10.1016/s0079-8169(08)x6002-7. ISBN 0-12-526740-1. MR 0719023. Zbl 0531.53051.
- Petersen, Peter (2016). Riemannian geometry. Graduate Texts in Mathematics. Vol. 171 (Third edition of 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. MR 3469435. Zbl 1417.53001.
External links
- Voitsekhovskii, M. I. (2001) [1994], "Hopf–Rinow theorem", Encyclopedia of Mathematics, EMS Press
- Derwent, John. "Hopf–Rinow theorem". MathWorld.
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