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:

\$\$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)

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