View source for Hopf-Rinow theorem

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480101.png" /> is a connected Riemannian space with distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480102.png" /> and a [[Levi-Civita connection|Levi-Civita connection]], then the following assertions are equivalent:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480103.png" /> is complete;

2) for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480104.png" /> the [[Exponential mapping|exponential mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480105.png" /> is defined on the whole tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480106.png" />;

3) every closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480107.png" /> that is bounded with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480108.png" /> is compact.

===Corollary:===
Any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480109.png" /> can be joined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801010.png" /> by a geodesic of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801011.png" />. This was established by H. Hopf and W. Rinow [[#References|[1]]].

A generalization of the Hopf–Rinow theorem (see [[#References|[4]]]) is: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801013.png" /> are two points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801014.png" />, then either there exists a curve joining them in a shortest way or there exists a geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801015.png" /> emanating from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801016.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801017.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801018.png" />; 2) if a sequence of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801019.png" /> does not have limit points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801020.png" />, then it does not have limit points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801021.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801022.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801023.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801024.png" /> contains the shortest connection between any two points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801025.png" />; 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801026.png" /> for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801027.png" />; and 5) the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801028.png" /> is finite and does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801029.png" />. Here the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801030.png" /> is not necessarily symmetric, and every point can be joined in a shortest possible (not necessarily unique) way to any point in a certain neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801031.png" />.

===Corollary:===
If there are no bounded rays in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801032.png" />, then every bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801033.png" /> is compact.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hopf,   W. Rinow,   "Ueber den Begriff der vollständigen differentialgeometrischen Flächen"  ''Comm. Math. Helv.'' , '''3'''  (1931)  pp. 209–225</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. de Rham,   "Sur la réducibilité d'un espace de Riemann"  ''Comm. Math. Helv.'' , '''26'''  (1952)  pp. 328–344</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Gromoll,   W. Klingenberg,   W. Meyer,   "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.E. Cohn-Vossen,   "Some problems of differential geometry in the large" , Moscow  (1959)  (In Russian)</TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801034.png" />. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801035.png" /> is called geodesically complete at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801036.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801037.png" /> is defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801038.png" />. The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801039.png" /> is geodesically complete if this holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801040.png" />. The Hopf–Rinow theorem also includes the statement that geodesic completeness is equivalent to geodesic completeness at one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801041.png" />.

A geodesic joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h04801043.png" /> and of minimal length is called a minimizing geodesic.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>