Hopf–Rinow theorem

<h2 id="statement">Statement</h2>
Let 
 
 
 
 (
 M
 ,
 g
 )
 
 
 {\displaystyle (M,g)}
 
 be a <a href="/facts/Connected_space/5CUTxfoA">connected</a> and smooth Riemannian manifold. Then the following statements are equivalent:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<ol><li>The <a href="/facts/Closed_set/upYnRTUj">closed</a> and <a href="/facts/Bounded_set/yCldjtoy">bounded</a> <a href="/facts/Subset/SJGERmbA">subsets</a> of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 are <a href="/facts/Compact_space/d0cgXJH7">compact</a>;</li>
<li>
 
 
 
 M
 
 
 {\displaystyle M}
 
 is a <a href="/facts/Complete_space/lsckmU8G">complete</a> <a href="/facts/Metric_space/gnBBRXtc">metric space</a>;</li>
<li>
 
 
 
 M
 
 
 {\displaystyle M}
 
 is geodesically complete; that is, for every 
 
 
 
 p
 ∈
 M
 ,
 
 
 {\displaystyle p\in M,}
 
 the <a href="/facts/Exponential_map_(Riemannian_geometry)/YfsKa30B">exponential map</a> expp is defined on the entire <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> 
 
 
 
 
 T
 
 p
 
 
 ⁡
 M
 .
 
 
 {\displaystyle \operatorname {T} _{p}M.}
 
</li></ol>
Furthermore, any one of the above implies that given any two points 
 
 
 
 p
 ,
 q
 ∈
 M
 ,
 
 
 {\displaystyle p,q\in M,}
 
 there exists a length minimizing <a href="/facts/Geodesic/fbWR6l80">geodesic</a> connecting these two points (geodesics are in general <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical points</a> for the <a href="/facts/Arc_length/QgTrmLxY">length</a> 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 <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a> (namely minimization of the length functional); the third deals with the nature of solutions to a certain system of <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a>.

<h2 id="variations-and-generalizations">Variations and generalizations</h2>
<ul><li>The Hopf–Rinow theorem is generalized to <a href="/facts/Length-metric_space/V0TslFne">length-metric spaces</a> the following way:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
<ul><li>If a <a href="/facts/Length-metric_space/V0TslFne">length-metric space</a> is <a href="/facts/Complete_space/lsckmU8G">complete</a> and <a href="/facts/Locally_compact/hesalT7q">locally compact</a> then any two points can be connected by a <a href="/facts/Geodesic/fbWR6l80">minimizing geodesic</a>, and any bounded <a href="/facts/Closed_set/upYnRTUj">closed set</a> is <a href="/facts/Compact_space/d0cgXJH7">compact</a>.</li></ul></li></ul>
In fact these properties characterize completeness for locally compact length-metric spaces.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
<ul><li>The theorem does not hold for infinite-dimensional manifolds. The unit sphere in a <a href="/facts/Separable_Hilbert_space/aDFMh4lV">separable Hilbert space</a> can be endowed with the structure of a <a href="/facts/Hilbert_manifold/b1n79WkP">Hilbert manifold</a> in such a way that antipodal points cannot be joined by a length-minimizing geodesic.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>The theorem also does not generalize to <a href="/facts/Lorentzian_manifold/VFXsVjqb">Lorentzian manifolds</a>: the <a href="/facts/Clifton%E2%80%93Pohl_torus/hP6zu85T">Clifton–Pohl torus</a> provides an example (diffeomorphic to the two-dimensional torus) that is compact but not complete.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Dmitri_Burago/1LEzHcBK">Burago, Dmitri</a>; <a href="/facts/Yuri_Burago/02IfQ5Fm">Burago, Yuri</a>; <a href="/facts/Sergei_Ivanov_(mathematician)/7LYJYFbx">Ivanov, Sergei</a> (2001). A course in metric geometry. <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>. Vol. 33. Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fgsm%2F033">10.1090/gsm/033</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-2129-6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1835418">1835418</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0981.51016">0981.51016</a>. (Erratum:  <a href="https://www.pdmi.ras.ru/~svivanov/papers/bbi-errata.pdf">[1]</a>)</li>
<li><a href="/facts/Martin_Bridson/q63MNtcW">Bridson, Martin R.</a>; <a href="/facts/Andr%C3%A9_Haefliger/VFEy0jaS">Haefliger, André</a> (1999). Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften. Vol. 319. Berlin: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-662-12494-9">10.1007/978-3-662-12494-9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-64324-9. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1744486">1744486</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0988.53001">0988.53001</a>.</li>
<li><a href="/facts/Manfredo_do_Carmo/5yxcMKQB">do Carmo, Manfredo Perdigão</a> (1992). Riemannian geometry. Mathematics: Theory & Applications. Translated from the second Portuguese edition by Francis Flaherty. Boston, MA: <a href="/facts/Birkh%C3%A4user/brvgnrnJ">Birkhäuser Boston, Inc.</a> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-3490-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1138207">1138207</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0752.53001">0752.53001</a>.</li>
<li><a href="/facts/Sylvestre_Gallot/DcsNcs2h">Gallot, Sylvestre</a>; <a href="/facts/Dominique_Hulin/FbtFlJPv">Hulin, Dominique</a>; Lafontaine, Jacques (2004). Riemannian geometry. Universitext (Third ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-18855-8">10.1007/978-3-642-18855-8</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-20493-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2088027">2088027</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1068.53001">1068.53001</a>.</li>
<li><a href="/facts/Mikhael_Gromov_(mathematician)/N0t5hhcI">Gromov, Misha</a> (1999). <a href="/facts/Metric_Structures_for_Riemannian_and_Non-Riemannian_Spaces/DWTDq672">Metric structures for Riemannian and non-Riemannian spaces</a>. Progress in Mathematics. Vol. 152. Translated by Bates, Sean Michael. With appendices by <a href="/facts/Mikhail_Katz/2Fy17Yyy">M. Katz</a>, <a href="/facts/Pierre_Pansu/IwA5NJJl">P. Pansu</a>, and <a href="/facts/Stephen_Semmes/9lN6kxP4">S. Semmes</a>. (Based on the 1981 French original ed.). Boston, MA: <a href="/facts/Birkh%C3%A4user/brvgnrnJ">Birkhäuser Boston, Inc.</a> <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-0-8176-4583-0">10.1007/978-0-8176-4583-0</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-3898-9. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1699320">1699320</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0953.53002">0953.53002</a>.</li>
<li><a href="/facts/J%C3%BCrgen_Jost/lRS3cQWE">Jost, Jürgen</a> (2017). Riemannian geometry and geometric analysis. Universitext (Seventh edition of 1995 original ed.). <a href="/facts/Springer,_Cham/TpTgApAh">Springer, Cham</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-319-61860-9">10.1007/978-3-319-61860-9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-61859-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3726907">3726907</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1380.53001">1380.53001</a>.</li>
<li><a href="/facts/Shoshichi_Kobayashi/P3rHgfGs">Kobayashi, Shoshichi</a>; <a href="/facts/Katsumi_Nomizu/Xhs8rgXm">Nomizu, Katsumi</a> (1963). <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of differential geometry. Volume I</a>. New York–London: <a href="/facts/John_Wiley_%26_Sons,_Inc./53g3LqJc">John Wiley & Sons, Inc.</a> <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0152974">0152974</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0119.37502">0119.37502</a>.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1999). Fundamentals of differential geometry. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 191. New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0541-8">10.1007/978-1-4612-0541-8</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98593-X. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1666820">1666820</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0932.53001">0932.53001</a>.</li>
<li><a href="/facts/Barrett_O%27Neill/mnbSvRIV">O'Neill, Barrett</a> (1983). Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics. Vol. 103. New York: <a href="/facts/Academic_Press/qxoKjzFy">Academic Press, Inc.</a> <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fs0079-8169%2808%29x6002-7">10.1016/s0079-8169(08)x6002-7</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-526740-1. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0719023">0719023</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0531.53051">0531.53051</a>.</li>
<li>Petersen, Peter (2016). Riemannian geometry. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 171 (Third edition of 1998 original ed.). <a href="/facts/Springer_Publishing/JfRev59b">Springer, Cham</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-319-26654-1">10.1007/978-3-319-26654-1</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-26652-7. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3469435">3469435</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1417.53001">1417.53001</a>.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Voitsekhovskii, M. I. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Hopf–Rinow_theorem">"Hopf–Rinow theorem"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Derwent, John. <a href="https://mathworld.wolfram.com/Hopf-RinowTheorem.html">"Hopf–Rinow theorem"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hopf, H.; Rinow, W. (1931). "Ueber den Begriff der vollständigen differentialgeometrischen Fläche". Commentarii Mathematici Helvetici. 3 (1): 209–225. doi:10.1007/BF01601813. <a href="/wiki/Commentarii_Mathematici_Helvetici" target="_blank">/wiki/Commentarii_Mathematici_Helvetici</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">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. - 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. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1138207" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1138207</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Bridson & Haefliger 1999, Proposition I.3.7; Gromov 1999, Section 1.B. - 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. <a href="https://doi.org/10.1007%2F978-3-662-12494-9" target="_blank">https://doi.org/10.1007%2F978-3-662-12494-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Burago, Burago & Ivanov 2001, Section 2.5.3. - 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:  [1]) <a href="https://doi.org/10.1090%2Fgsm%2F033" target="_blank">https://doi.org/10.1090%2Fgsm%2F033</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Lang 1999, pp. 226–227. - 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. <a href="https://doi.org/10.1007%2F978-1-4612-0541-8" target="_blank">https://doi.org/10.1007%2F978-1-4612-0541-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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 <a href="/wiki/The_Bulletin_of_the_London_Mathematical_Society" target="_blank">/wiki/The_Bulletin_of_the_London_Mathematical_Society</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Gallot, Hulin & Lafontaine 2004, Section 2.D.4; O'Neill 1983, p. 193. - 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. <a href="https://doi.org/10.1007%2F978-3-642-18855-8" target="_blank">https://doi.org/10.1007%2F978-3-642-18855-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Hopf–Rinow theorem open-in-new

Hopf–Rinow theorem