Bianchi identity
From Encyclopedia of Mathematics
A relation connecting the components of the covariant derivatives of the curvature tensor 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
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