Editing Hopf–Rinow theorem (section)

==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}}