Bianchi identity

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


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


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


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


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


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