Hopf–Rinow theorem#Variations and generalizations

Let $(M,\; g)$ be a connected and smooth Riemannian manifold. Then the following statements are equivalent:{{sfnm|1a1=do Carmo|1y=1992|1loc=Chapter 7|2a1=Gallot|2a2=Hulin|2a3=Lafontaine|2y=2004|2loc=Section 2.C.5|3a1=Jost|3y=2017|3loc=Section 1.7|4a1=Kobayashi|4a2=Nomizu|4y=1963|4loc=Section IV.4|5a1=Lang|5y=1999|5loc=Section VIII.6|6a1=O'Neill|6y=1983|6loc=Theorem 5.21 and Proposition 5.22|7a1=Petersen|7y=2016|7loc=Section 5.7.1}}

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 $p\; \backslash in\; M,$ the exponential map exp_p is defined on the entire tangent space $\backslash operatorname\{T\}\_p\; M.$

Furthermore, any one of the above implies that given any two points $p,\; q\; \backslash in\; M,$ 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.

• The Hopf–Rinow theorem is generalized to length-metric spaces the following way:{{sfnm|1a1=Bridson|1a2=Haefliger|1y=1999|1loc=Proposition I.3.7|2a1=Gromov|2y=1999|2loc=Section 1.B}}
• 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.{{sfnm|1a1=Burago|1a2=Burago|1a3=Ivanov|1y=2001|1loc=Section 2.5.3}}

• 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.{{sfnm|1a1=Lang|1y=1999|1pp=226–227}} It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not.{{Citation|last1=Atkin|first1=C. J.|title=The Hopf–Rinow theorem is false in infinite dimensions|mr=0400283|year=1975|journal=The Bulletin of the London Mathematical Society|volume=7|issue=3|pages=261–266|doi=10.1112/blms/7.3.261}}
• 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.{{sfnm|1a1=Gallot|1a2=Hulin|1a3=Lafontaine|1y=2004|1loc=Section 2.D.4|2a1=O'Neill|2y=1983|2p=193}}

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• {{cite book|last1=Jost|first1=Jürgen|title=Riemannian geometry and geometric analysis|series=Universitext|author-link1=Jürgen Jost|edition=Seventh edition of 1995 original|publisher=Springer, Cham|year=2017|isbn=978-3-319-61859-3|mr=3726907|doi=10.1007/978-3-319-61860-9|zbl=1380.53001}}
• {{cite book|author-link1=Shoshichi Kobayashi|author-link2=Katsumi Nomizu|last1=Kobayashi|first1=Shoshichi|last2=Nomizu|title-link=Foundations of Differential Geometry|first2=Katsumi|title=Foundations of differential geometry. Volume I |publisher=John Wiley & Sons, Inc.|location=New York–London|year=1963|mr=0152974|zbl=0119.37502}}
• {{cite book|mr=1666820|last1=Lang|first1=Serge|title=Fundamentals of differential geometry|series=Graduate Texts in Mathematics|volume=191|publisher=Springer-Verlag|location=New York|year=1999|isbn=0-387-98593-X|doi=10.1007/978-1-4612-0541-8|author-link1=Serge Lang|zbl=0932.53001}}
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• {{springer|id=H/h048010|title=Hopf–Rinow theorem|first=M. I. |last=Voitsekhovskii}}
• {{mathworld|Hopf-RinowTheorem|author=Derwent, John}}

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Category:Metric geometry

Category:Theorems in Riemannian geometry