Bianchi identity

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)