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)

Comments

The first Ricci identity is usually called the first Bianchi identity in the West, cf. also Bianchi identity.

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a3]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)