# Ricci identity

An identity expressing one of the properties of the Riemann tensor $R _ {ij,k} ^ {l}$( or $R _ {ij,kl}$):

$$R _ {ij,k} ^ {l} + R _ {jk,i} ^ {l} + R _ {ki,j} ^ {l} = 0 .$$

For a covariant tensor $R _ {ij,kl}$ the identity is of the form

$$R _ {ij,kl} + R _ {jk,il} + R _ {ki,jl} = 0 ,$$

i.e. cycling over the three first indices yields zero.

An identity which should be satisfied by the covariant derivatives of second order with respect to the metric tensor $g _ {ij}$ of a Riemannian space $V _ {n}$, which differ only by the order of differentiation. If $\lambda _ {i}$ is a tensor of valency 1 and $\lambda _ {i,jk}$ is the covariant derivative of second order with respect to $x ^ {j}$ and $x ^ {k}$ relative to the tensor $g _ {ij}$, then the Ricci identity takes the form

$$\lambda _ {i,jk} - \lambda _ {i,kj} = \lambda _ {l} R _ {ij,k} ^ {l} ,$$

where $R _ {ij,k} ^ {l}$ is the Riemann curvature tensor determined by the metric tensor $g _ {ij}$ of the space $V _ {n}$( in other words, an alternating second absolute derivative of the tensor field $\lambda _ {i}$ in the metric $g _ {ij}$ is expressed in terms of the Riemann tensor and the components of $\lambda _ {i}$).

For a covariant tensor $a _ {ij}$ of valency 2 the Ricci identity has the form

$$a _ {ij,kl} - a _ {ij,lk} = \ a _ {ih} R _ {jk,l} ^ {h} + a _ {h j } R _ {ik,l} ^ {h} .$$

In general, for a covariant tensor $a _ {r _ {1} \dots r _ {m} }$ of valency $m$ the identity has the form

$$a _ {r _ {1} \dots r _ {m} , k l } - a _ {r _ {1} \dots r _ {m} , l k } =$$

$$= \ \sum _ \alpha ^ { {1 } \dots m } a _ {r _ {1} \dots r _ {\alpha - 1 } h r _ {\alpha + 1 } \dots r _ {m} } R _ {r _ \alpha k l } ^ {h} .$$

Similar identities can be written for contravariant and mixed tensors in $V _ {n}$. The Ricci identity is used, e.g., in constructions of the geometry of subspaces in $V _ {n}$ as an integrability condition for the principal variational equations from which Gauss' equations and the Peterson–Codazzi equations for subspaces of $V _ {n}$ are derived.

The identity was established by G. Ricci (see [1]).

#### References

 [1] G. Ricci, T. Levi-Civita, "Méthodes de calcul différentiel absolu et leurs applications" Math. Ann. , 54 (1901) pp. 125–201 [2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) [3] L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)