# Bianchi identity

From Encyclopedia of Mathematics

A relation connecting the components of the covariant derivatives of the curvature tensor of a Riemannian space:

where . First established by L. Bianchi [1] in 1902.

#### References

[1] | L. Bianchi, "Lezioni di geometria differenziale" , 1–2 , Zanichelli , Bologna (1923–1927) |

#### Comments

Here denotes of course the covariant derivative of with respect to the -th coordinate.

The identity described above is often called the second Bianchi identity. The first Bianchi identity is then given by

(see [a1], [a2]). Generalized versions of these identities for curvature forms and curvature tensors of connections with torsion are given in [a2].

#### References

[a1] | N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965) |

[a2] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) |

**How to Cite This Entry:**

Bianchi identity.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bianchi_identity&oldid=13332

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article