Bianchi identity
From Encyclopedia of Mathematics
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A relation connecting the components of the covariant derivatives of the curvature tensor 
of a Riemannian space:
 |
where 
. First established by L. Bianchi [1] in 1902.
References
| [1] | L. Bianchi, "Lezioni di geometria differenziale" , 1–2 , Zanichelli , Bologna (1923–1927) |
Comments
Here 
denotes of course the covariant derivative of 
with respect to the 
-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].
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=13332
Bianchi identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bianchi_identity&oldid=13332
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article