Riemann tensor

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)

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)