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Difference between revisions of "Hopf-Rinow theorem"

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If $M$ is a connected Riemannian space with distance function $\rho$ and a [[Levi-Civita connection|Levi-Civita connection]], then the following assertions are equivalent:
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If $M$ is a connected [[Riemannian space]] with [[Riemannian metric]] $\rho$ and a [[Levi-Civita connection]], then the following assertions are equivalent:
  
1) $M$ is complete;
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1) $M$ is a [[complete Riemannian space]];
  
2) for every point $p\in M$ the [[Exponential mapping|exponential mapping]] $\exp_p$ is defined on the whole tangent space $M_p$;
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2) for every point $p\in M$ the [[exponential mapping]] $\exp_p$ is defined on the whole tangent space $M_p$;
  
 
3) every closed set $A\subset M$ that is bounded with respect to $\rho$ is compact.
 
3) every closed set $A\subset M$ that is bounded with respect to $\rho$ is compact.
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====References====
 
====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>
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<table>
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<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>
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<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>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR>
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<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>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR>
 +
</table>

Latest revision as of 22:13, 7 March 2016

If $M$ is a connected Riemannian space with Riemannian metric $\rho$ and a Levi-Civita connection, then the following assertions are equivalent:

1) $M$ is a complete Riemannian space;

2) for every point $p\in M$ the exponential mapping $\exp_p$ is defined on the whole tangent space $M_p$;

3) every closed set $A\subset M$ that is bounded with respect to $\rho$ is compact.

Corollary:

Any two points $p,q\in M$ can be joined in $M$ by a geodesic of length $\rho(p,q)$. This was established by H. Hopf and W. Rinow [1].

A generalization of the Hopf–Rinow theorem (see [4]) is: If $p$ and $q$ are two points in $M$, then either there exists a curve joining them in a shortest way or there exists a geodesic $L$ emanating from $p$ with the following properties: 1) $L$ is homeomorphic to $0\leq t<1$; 2) if a sequence of points on $L$ does not have limit points on $L$, then it does not have limit points in $M$, that is, $L$ is closed in $M$; 3) $L$ contains the shortest connection between any two points on $L$; 4) $\rho(p,x)+\rho(x,q)=\rho(p,q)$ for every point $x\in L$; and 5) the length of $L$ is finite and does not exceed $\rho(p,q)$. Here the function $\rho(p,q)$ 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 $U_p$.

Corollary:

If there are no bounded rays in $M$, then every bounded set in $M$ is compact.

References

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


Comments

Let $p\in M$. The manifold $M$ is called geodesically complete at $p$ if $\exp_p$ is defined on all of $T_pM$. The manifold $M$ is geodesically complete if this holds for all $p$. The Hopf–Rinow theorem also includes the statement that geodesic completeness is equivalent to geodesic completeness at one $p\in M$.

A geodesic joining $p$ and $q$ and of minimal length is called a minimizing geodesic.

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
How to Cite This Entry:
Hopf-Rinow theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf-Rinow_theorem&oldid=32950
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article