Editing Hopf–Rinow theorem (section)

==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.