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| − | 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: | + | {{TEX|done}} |
| | + | 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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| − | 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048010/h0480103.png" /> is complete; | + | 1) $M$ is complete; |
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| − | 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" />; | + | 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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| − | 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. | + | 3) every closed set $A\subset M$ that is bounded with respect to $\rho$ is compact. |
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| | ===Corollary:=== | | ===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]]]. | + | 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 [[#References|[1]]]. |
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| − | 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" />. | + | A generalization of the Hopf–Rinow theorem (see [[#References|[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 $U_p$. |
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| | ===Corollary:=== | | ===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. | + | If there are no bounded rays in $M$, then every bounded set in $M$ is compact. |
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| | ====References==== | | ====References==== |
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| | ====Comments==== | | ====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" />. | + | 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 $p\in M$. |
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| − | 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. | + | A geodesic joining $p$ and $q$ and of minimal length is called a minimizing geodesic. |
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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> | | <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 06:33, 15 August 2014
If $M$ is a connected Riemannian space with distance function $\rho$ and a Levi-Civita connection, then the following assertions are equivalent:
1) $M$ is complete;
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) |
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=22591
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article