View source for Hopf–Rinow theorem

{{Short description|Gives equivalent statements about the geodesic completeness of Riemannian manifolds}}
'''Hopf–Rinow theorem''' is a set of statements about the [[Geodesic manifold|geodesic completeness]] of [[Riemannian manifold]]s. It is named after [[Heinz Hopf]] and his student [[Willi Rinow]], who published it in 1931.<ref>{{cite journal|last1=Hopf|first1=H.|last2=Rinow|first2=W.|title=Ueber den Begriff der vollständigen differentialgeometrischen Fläche|journal=[[Commentarii Mathematici Helvetici]]|volume=3|year=1931|issue=1|pages=209–225|doi=10.1007/BF01601813}}</ref> [[Stefan Cohn-Vossen]] extended part of the Hopf–Rinow theorem to the context of certain types of [[metric space]]s.

==Statement==
Let <math>(M, g)</math> be a [[Connected space|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}}

# The [[Closed set|closed]] and [[Bounded set|bounded]] [[subset]]s of <math>M</math> are [[Compact space|compact]];
# <math>M</math> is  a [[Complete space|complete]] [[metric space]];
# <math>M</math> is geodesically complete; that is, for every <math>p \in M,</math> the [[Exponential map (Riemannian geometry)|exponential map]] exp<sub>''p''</sub> is defined on the entire [[tangent space]] <math>\operatorname{T}_p M.</math>

Furthermore, any one of the above implies that given any two points <math>p, q \in M,</math> there exists a length minimizing [[geodesic]] connecting these two points (geodesics are in general [[Critical point (mathematics)|critical points]] for the [[Arc length|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 equation]]s.

==Variations and generalizations==
* The Hopf–Rinow theorem is generalized to [[length-metric space]]s 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 space|complete]] and [[locally compact]] then any two points can be connected by a [[Geodesic|minimizing geodesic]], and any bounded [[closed set]] is [[Compact space|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.<ref>{{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}}</ref>
*The theorem also does not generalize to [[Lorentzian manifold]]s: 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}}

==Notes==
{{reflist}}

==References==
{{refbegin}}
* {{wikicite|ref={{sfnRef|Burago|Burago|Ivanov|2001}}|reference={{cite book|mr=1835418|zbl=0981.51016|last1=Burago|first1=Dmitri|last2=Burago|first2=Yuri|last3=Ivanov|first3=Sergei|title=A course in metric geometry|series=[[Graduate Studies in Mathematics]]|volume=33|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2001|isbn=0-8218-2129-6|author-link1=Dmitri Burago|author-link2=Yuri Burago|author-link3=Sergei Ivanov (mathematician)|doi=10.1090/gsm/033|ref=none}} {{erratum|https://www.pdmi.ras.ru/~svivanov/papers/bbi-errata.pdf|checked=yes}}}}
* {{cite book|mr=1744486|last1=Bridson|first1=Martin R.|last2=Haefliger|first2=André|title=Metric spaces of non-positive curvature|series=Grundlehren der mathematischen Wissenschaften|volume=319|publisher=[[Springer-Verlag]]|location=Berlin|year=1999|isbn=3-540-64324-9|doi=10.1007/978-3-662-12494-9|author-link1=Martin Bridson|author-link2=André Haefliger|zbl=0988.53001}}
* {{cite book|last=do Carmo|first=Manfredo Perdigão|authorlink=Manfredo do Carmo|title=Riemannian geometry|series= Mathematics: Theory & Applications|year=1992|isbn=0-8176-3490-8|location=Boston, MA|publisher=[[Birkhäuser|Birkhäuser Boston, Inc.]]|others=Translated from the second Portuguese edition by Francis Flaherty|zbl=0752.53001|mr=1138207}}
* {{cite book|last1=Gallot|first1=Sylvestre|author-link1=Sylvestre Gallot|last2=Hulin|first2=Dominique|author-link2=Dominique Hulin|last3=Lafontaine|first3=Jacques|title=Riemannian geometry|year=2004|edition=Third|series=Universitext|publisher=[[Springer-Verlag]]|mr=2088027|isbn=3-540-20493-8|doi=10.1007/978-3-642-18855-8|zbl=1068.53001}}
* {{cite book|last1=Gromov|first1=Misha|title-link=Metric Structures for Riemannian and Non-Riemannian Spaces|title=Metric structures for Riemannian and non-Riemannian spaces|edition=Based on the 1981 French original|series=Progress in Mathematics|volume=152|publisher=[[Birkhäuser|Birkhäuser Boston, Inc.]]|location=Boston, MA|year=1999|isbn=0-8176-3898-9|mr=1699320|translator-last1=Bates|translator-first1=Sean Michael|others=With appendices by [[Mikhail Katz|M. Katz]], [[Pierre Pansu|P. Pansu]], and [[Stephen Semmes|S. Semmes]].|doi=10.1007/978-0-8176-4583-0|zbl=0953.53002|author-link1=Mikhael Gromov (mathematician)}}
* {{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}}
* {{cite book|last1=O'Neill|first1=Barrett|author-link1=Barrett O'Neill|title=Semi-Riemannian geometry. With applications to relativity|series=Pure and Applied Mathematics|volume=103|publisher=[[Academic Press|Academic Press, Inc.]]|location=New York|year=1983|isbn=0-12-526740-1|mr=0719023|zbl=0531.53051|doi=10.1016/s0079-8169(08)x6002-7}}
* {{cite book|last1=Petersen|first1=Peter|title=Riemannian geometry|edition=Third edition of 1998 original|series=[[Graduate Texts in Mathematics]]|volume=171|publisher=[[Springer Publishing|Springer, Cham]]|year=2016|isbn=978-3-319-26652-7|mr=3469435|doi=10.1007/978-3-319-26654-1|zbl=1417.53001}}
{{refend}}

==External links==
* {{springer|id=H/h048010|title=Hopf–Rinow theorem|first=M. I. |last=Voitsekhovskii}}
* {{mathworld|Hopf-RinowTheorem|author=Derwent, John}}
{{Riemannian geometry}}
{{Manifolds}}

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[[Category:Metric geometry]]
[[Category:Theorems in Riemannian geometry]]