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''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|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|Gaussian curvature]]).

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

====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>

====Comments====

====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>