# Bianchi identity

From Encyclopedia of Mathematics

A relation connecting the components of the covariant derivatives of the curvature tensor $R_{ijk}^h$ of a Riemannian space:

$$R_{ijk,l}^h+R_{ikl,j}^h+R_{ilj,k}^h=0,$$

where $h,i,j,k,l=1,\dots,n$. First established by L. Bianchi [1] in 1902.

#### References

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

#### Comments

Here $R_{ijk,l}^h$ denotes of course the covariant derivative of $R_{ijk}^h$ with respect to the $l$-th coordinate.

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

$$R_{jkl}^i+R_{klj}^i+R_{ljk}^i=0$$

(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=34093

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