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Difference between revisions of "Complete Riemannian space"

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A Riemannian space with its internal distance function $\rho$ that is complete as a metric space with metric $\rho$.
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A [[Riemannian space]] that is a [[complete metric space]] with respect to the [[Riemannian 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|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.
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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]] $\exp_p$ 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.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Existenz kürzester Wege"  ''Compos. Math.'' , '''3'''  (1936)  pp. 441–452</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Existenz kürzester Wege"  ''Compos. Math.'' , '''3'''  (1936)  pp. 441–452</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>
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</table>

Latest revision as of 22:10, 7 March 2016

A Riemannian space that is a complete metric space with respect to the Riemannian 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=37709
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article