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''Riemann curvature tensor''
 
''Riemann curvature tensor''
  
A four-valent tensor that is studied in the theory of curvature of spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820701.png" /> be a space with an affine connection and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820702.png" /> be the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]]) of the connection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820703.png" />. The components (coordinates) of the Riemann tensor, which is once contravariant and three times covariant, take the form
+
A four-valent tensor that is studied in the theory of curvature of spaces. Let $  L _ {n} $
 +
be a space with an affine connection and let $  \Gamma _ {ij}  ^ {k} $
 +
be the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]]) of the connection of $  L _ {n} $.  
 +
The components (coordinates) of the Riemann tensor, which is once contravariant and three times covariant, take the form
 +
 
 +
$$
 +
R _ {lki}  ^ {q}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820704.png" /></td> </tr></table>
+
\frac{\partial  \Gamma _ {li}  ^ {q} }{\partial  x  ^ {k} }
 +
-  
 +
\frac{\partial  \Gamma _ {ki}  ^ {q} }{\partial  x  ^ {l} }
 +
- \Gamma _ {lp}  ^ {q} \Gamma _ {ki}  ^ {p} +
 +
\Gamma _ {kp}  ^ {q} \Gamma _ {li}  ^ {p} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820705.png" /></td> </tr></table>
+
$$
 +
l, k, i, q  = 1 \dots n,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820706.png" /> is the symbol of differentiation with respect to the space coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820708.png" />. In a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r0820709.png" /> with a metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207010.png" />, in addition to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207011.png" /> the four times covariant Riemann tensor obtained by lowering the upper index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207012.png" /> using the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207013.png" /> is also studied
+
where $  \partial  / \partial  x _ {k} $
 +
is the symbol of differentiation with respect to the space coordinate $  x  ^ {k} $,  
 +
$  k = 1 \dots n $.  
 +
In a Riemannian space $  V _ {n} $
 +
with a metric tensor $  g _ {ij} $,  
 +
in addition to the tensor $  R _ {lki}  ^ {q} $
 +
the four times covariant Riemann tensor obtained by lowering the upper index $  q $
 +
using the metric tensor $  g _ {ij} $
 +
is also studied
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207014.png" /></td> </tr></table>
+
$$
 +
R _ {lki}  ^ {q} g _ {j}  = \
 +
R _ {lkij\ } \equiv
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207015.png" /></td> </tr></table>
+
$$
 +
\equiv \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207016.png" /></td> </tr></table>
+
\frac{1}{2}
 +
\left (
 +
\frac{\partial  ^ {2} g _ {lj} }{\partial  x
 +
^ {k} \partial  x  ^ {i} }
 +
-  
 +
\frac{\partial  ^ {2} g _ {li}  }{\partial  x  ^ {k} \partial  x  ^ {j} }
 +
-
 +
\frac{\partial  ^ {2} g _ {kj} }{\partial  x  ^ {l} \partial  x  ^ {i} }
 +
+
 +
\frac{\partial  ^ {2}
 +
g _ {ki} }{\partial  x  ^ {l} \partial  x  ^ {j} }
 +
\right ) +
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207017.png" /> since the Riemannian connection (without torsion) is considered on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207018.png" />. In an arbitrary space with an affine connection without torsion the coordinates of the Riemann tensor satisfy the first Bianchi identity
+
$$
 +
+
 +
g _ {pq} ( \Gamma _ {lj}  ^ {p} \Gamma _ {ki}  ^ {q} - \Gamma _ {kj}  ^ {p} \Gamma _ {li}  ^ {q} ).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207019.png" /></td> </tr></table>
+
Here  $  \Gamma _ {ij}  ^ {k} = \Gamma _ {ji}  ^ {k} $
 +
since the Riemannian connection (without torsion) is considered on  $  V _ {n} $.  
 +
In an arbitrary space with an affine connection without torsion the coordinates of the Riemann tensor satisfy the first Bianchi identity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207020.png" /></td> </tr></table>
+
$$
 +
R _ {lki}  ^ {q} + R _ {kil}  ^ {q} + R _ {ilk}  ^ {q}  = 0,
 +
$$
 +
 
 +
$$
 +
R _ {lkij} + R _ {kilj} + R _ {ilkj}  = 0,
 +
$$
  
 
i.e. the cyclic sum with respect to the first three subscripts is zero.
 
i.e. the cyclic sum with respect to the first three subscripts is zero.
Line 25: Line 85:
 
The Riemann tensor possesses the following properties:
 
The Riemann tensor possesses the following properties:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207021.png" />;
+
1) $  R _ {lkij} = R _ {ijlk} $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207022.png" />;
+
2) $  R _ {ilk}  ^ {q} = - R _ {ikl}  ^ {q} $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207024.png" />;
+
3) $  R _ {lkij} = - R _ {klij} $,  
 +
$  R _ {lkij} = - R _ {lkji} $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207026.png" />, if both subscripts of one pair are identical, then the corresponding coordinate equals zero: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207027.png" />;
+
4) $  R _ {aaij} = 0 $,  
 +
$  R _ {lkbb} = 0 $,  
 +
if both subscripts of one pair are identical, then the corresponding coordinate equals zero: $  R _ {aai}  ^ {q} = 0 $;
  
 
5) the second Bianchi identity is applicable to the absolute derivatives of the Riemann tensor:
 
5) the second Bianchi identity is applicable to the absolute derivatives of the Riemann tensor:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207028.png" /></td> </tr></table>
+
$$
 +
\nabla _ {m} R _ {kli}  ^ {q} + \nabla _ {k} R _ {jmi}  ^ {q} + \nabla _ {l} R _ {mki}  ^ {q} = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207029.png" /> is the symbol for covariant differentiation in the direction of the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207030.png" />. The same identity is applicable to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207031.png" />.
+
where $  \nabla _ {m} $
 +
is the symbol for covariant differentiation in the direction of the coordinate $  x  ^ {m} $.  
 +
The same identity is applicable to the tensor $  R _ {lkij} $.
  
A Riemann tensor has, in all, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207032.png" /> coordinates, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207033.png" /> being the dimension of the space, among which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207034.png" /> are essential. Between the latter no additional dependencies result from the properties listed above.
+
A Riemann tensor has, in all, $  n  ^ {4} $
 +
coordinates, $  n $
 +
being the dimension of the space, among which $  n  ^ {2} ( n  ^ {2} - 1)/12 $
 +
are essential. Between the latter no additional dependencies result from the properties listed above.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207035.png" /> the Riemann tensor has one essential coordinate, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207036.png" />; it forms part of the definition of the intrinsic, or Riemannian, curvature of the surface: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082070/r08207037.png" /> (see [[Gaussian curvature|Gaussian curvature]]).
+
When $  n= 2 $
 +
the Riemann tensor has one essential coordinate, $  R _ {1212} $;  
 +
it forms part of the definition of the intrinsic, or Riemannian, curvature of the surface: $  K = R _ {1212} / \mathop{\rm det}  g _ {ij} $(
 +
see [[Gaussian curvature|Gaussian curvature]]).
  
 
The Riemann tensor was defined by B. Riemann in 1861 (published in 1876).
 
The Riemann tensor was defined by B. Riemann in 1861 (published in 1876).
Line 47: Line 120:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1949)</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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1949)</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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Schouten,  D.J. Struik,  "Einführung in die neueren Methoden der Differentialgeometrie" , '''2''' , Noordhoff  (1924)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.P. Eisenhart,  "An introduction to differential geometry with the use of the tensor calculus" , Princeton Univ. Press  (1947)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.A. Schouten,  "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Schouten,  D.J. Struik,  "Einführung in die neueren Methoden der Differentialgeometrie" , '''2''' , Noordhoff  (1924)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.P. Eisenhart,  "An introduction to differential geometry with the use of the tensor calculus" , Princeton Univ. Press  (1947)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.A. Schouten,  "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  (Translated from German)</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


Riemann curvature tensor

A four-valent tensor that is studied in the theory of curvature of spaces. Let $ L _ {n} $ be a space with an affine connection and let $ \Gamma _ {ij} ^ {k} $ be the Christoffel symbols (cf. Christoffel symbol) of the connection of $ L _ {n} $. The components (coordinates) of the Riemann tensor, which is once contravariant and three times covariant, take the form

$$ R _ {lki} ^ {q} = \ \frac{\partial \Gamma _ {li} ^ {q} }{\partial x ^ {k} } - \frac{\partial \Gamma _ {ki} ^ {q} }{\partial x ^ {l} } - \Gamma _ {lp} ^ {q} \Gamma _ {ki} ^ {p} + \Gamma _ {kp} ^ {q} \Gamma _ {li} ^ {p} , $$

$$ l, k, i, q = 1 \dots n, $$

where $ \partial / \partial x _ {k} $ is the symbol of differentiation with respect to the space coordinate $ x ^ {k} $, $ k = 1 \dots n $. In a Riemannian space $ V _ {n} $ with a metric tensor $ g _ {ij} $, in addition to the tensor $ R _ {lki} ^ {q} $ the four times covariant Riemann tensor obtained by lowering the upper index $ q $ using the metric tensor $ g _ {ij} $ is also studied

$$ R _ {lki} ^ {q} g _ {j} = \ R _ {lkij\ } \equiv $$

$$ \equiv \ \frac{1}{2} \left ( \frac{\partial ^ {2} g _ {lj} }{\partial x ^ {k} \partial x ^ {i} } - \frac{\partial ^ {2} g _ {li} }{\partial x ^ {k} \partial x ^ {j} } - \frac{\partial ^ {2} g _ {kj} }{\partial x ^ {l} \partial x ^ {i} } + \frac{\partial ^ {2} g _ {ki} }{\partial x ^ {l} \partial x ^ {j} } \right ) + $$

$$ + g _ {pq} ( \Gamma _ {lj} ^ {p} \Gamma _ {ki} ^ {q} - \Gamma _ {kj} ^ {p} \Gamma _ {li} ^ {q} ). $$

Here $ \Gamma _ {ij} ^ {k} = \Gamma _ {ji} ^ {k} $ since the Riemannian connection (without torsion) is considered on $ V _ {n} $. In an arbitrary space with an affine connection without torsion the coordinates of the Riemann tensor satisfy the first Bianchi identity

$$ R _ {lki} ^ {q} + R _ {kil} ^ {q} + R _ {ilk} ^ {q} = 0, $$

$$ R _ {lkij} + R _ {kilj} + R _ {ilkj} = 0, $$

i.e. the cyclic sum with respect to the first three subscripts is zero.

The Riemann tensor possesses the following properties:

1) $ R _ {lkij} = R _ {ijlk} $;

2) $ R _ {ilk} ^ {q} = - R _ {ikl} ^ {q} $;

3) $ R _ {lkij} = - R _ {klij} $, $ R _ {lkij} = - R _ {lkji} $;

4) $ R _ {aaij} = 0 $, $ R _ {lkbb} = 0 $, if both subscripts of one pair are identical, then the corresponding coordinate equals zero: $ R _ {aai} ^ {q} = 0 $;

5) the second Bianchi identity is applicable to the absolute derivatives of the Riemann tensor:

$$ \nabla _ {m} R _ {kli} ^ {q} + \nabla _ {k} R _ {jmi} ^ {q} + \nabla _ {l} R _ {mki} ^ {q} = 0, $$

where $ \nabla _ {m} $ is the symbol for covariant differentiation in the direction of the coordinate $ x ^ {m} $. The same identity is applicable to the tensor $ R _ {lkij} $.

A Riemann tensor has, in all, $ n ^ {4} $ coordinates, $ n $ being the dimension of the space, among which $ n ^ {2} ( n ^ {2} - 1)/12 $ are essential. Between the latter no additional dependencies result from the properties listed above.

When $ n= 2 $ the Riemann tensor has one essential coordinate, $ R _ {1212} $; it forms part of the definition of the intrinsic, or Riemannian, curvature of the surface: $ K = R _ {1212} / \mathop{\rm det} g _ {ij} $( see Gaussian curvature).

The Riemann tensor was defined by B. Riemann in 1861 (published in 1876).

References

[1] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[2] L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)
[3] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)

Comments

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1969)
[a2] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a3] J.A. Schouten, D.J. Struik, "Einführung in die neueren Methoden der Differentialgeometrie" , 2 , Noordhoff (1924)
[a4] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
[a5] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a6] L.P. Eisenhart, "An introduction to differential geometry with the use of the tensor calculus" , Princeton Univ. Press (1947)
[a7] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)
How to Cite This Entry:
Riemann tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_tensor&oldid=48554
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article