# 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)