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A Riemannian space with its internal distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238501.png" /> that is complete as a metric space with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238502.png" />.
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A Riemannian space with its internal distance function $\rho$ that is complete as a metric space with metric $\rho$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238503.png" /> be a connected Riemannian space with its Levi-Civita connection, then the following three assertions are equivalent: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238504.png" /> is complete; b) for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238505.png" /> the [[Exponential mapping|exponential mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238506.png" /> is defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238507.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238508.png" /> is the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c0238509.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385010.png" />); and c) every closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385011.png" /> that is bounded with respect to the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385012.png" /> is compact (the Hopf–Rinow theorem). Consequences are: Any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385013.png" /> in a complete Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385014.png" /> can be joined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385015.png" /> by a geodesic of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385016.png" />; any geodesic is indefinitely extendable.
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Let $M$ be a connected Riemannian space with its Levi-Civita connection, then the following three assertions are equivalent: a) $M$ is complete; b) for each point $p\in M$ the [[Exponential mapping|exponential mapping]] $\exp_p$ is defined on all of $M_p$ (where $M_p$ is the tangent space to $M$ at $p$); and c) every closed set $A\subset M$ that is bounded with respect to the distance $\rho$ is compact (the Hopf–Rinow theorem). Consequences are: Any two points $p,q$ in a complete Riemannian space $M$ can be joined in $M$ by a geodesic of length $\rho(p,q)$; any geodesic is indefinitely extendable.
  
 
There is a generalization [[#References|[2]]] of this theorem to the case of a space with a non-symmetric distance function.
 
There is a generalization [[#References|[2]]] of this theorem to the case of a space with a non-symmetric distance function.
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====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385017.png" /> be a point of the Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385018.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385019.png" /> is called geodesically complete at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385021.png" /> is defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385023.png" /> is geodesically complete if this is the case at all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385024.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385025.png" /> to be complete (or, equivalently, geodesically complete) it suffices that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023850/c02385026.png" /> is geodesically complete at one point.
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Let $p$ be a point of the Riemannian manifold $M$. Then $M$ is called geodesically complete at $p$ if $\exp_p$ is defined on all of $T_pM$, and $M$ is geodesically complete if this is the case at all $p$. For $M$ to be complete (or, equivalently, geodesically complete) it suffices that $M$ is geodesically complete at one point.
  
 
====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>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>

Revision as of 15:03, 14 October 2014

A Riemannian space with its internal distance function $\rho$ that is complete as a metric space with metric $\rho$.

Let $M$ be a connected Riemannian space with its Levi-Civita connection, then the following three assertions are equivalent: a) $M$ is complete; b) for each point $p\in M$ the exponential mapping $\exp_p$ is defined on all of $M_p$ (where $M_p$ is the tangent space to $M$ at $p$); and c) every closed set $A\subset M$ that is bounded with respect to the distance $\rho$ is compact (the Hopf–Rinow theorem). Consequences are: Any two points $p,q$ in a complete Riemannian space $M$ can be joined in $M$ by a geodesic of length $\rho(p,q)$; any geodesic is indefinitely extendable.

There is a generalization [2] of this theorem to the case of a space with a non-symmetric distance function.

References

[1] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2] S.E. Cohn-Vossen, "Existenz kürzester Wege" Compos. Math. , 3 (1936) pp. 441–452


Comments

Let $p$ be a point of the Riemannian manifold $M$. Then $M$ is called geodesically complete at $p$ if $\exp_p$ is defined on all of $T_pM$, and $M$ is geodesically complete if this is the case at all $p$. For $M$ to be complete (or, equivalently, geodesically complete) it suffices that $M$ is geodesically complete at one point.

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
How to Cite This Entry:
Complete Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_Riemannian_space&oldid=11489
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article