Difference between revisions of "Bianchi identity"
From Encyclopedia of Mathematics
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| − | A relation connecting the components of the covariant derivatives of the [[Curvature tensor|curvature tensor]] | + | {{TEX|done}} |
| + | A relation connecting the components of the covariant derivatives of the [[Curvature tensor|curvature tensor]] $R_{ijk}^h$ of a Riemannian space: | ||
| − | + | $$R_{ijk,l}^h+R_{ikl,j}^h+R_{ilj,k}^h=0,$$ | |
| − | where | + | where $h,i,j,k,l=1,\dots,n$. First established by L. Bianchi [[#References|[1]]] in 1902. |
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | Here | + | 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 | 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 [[#References|[a1]]], [[#References|[a2]]]). Generalized versions of these identities for curvature forms and curvature tensors of connections with torsion are given in [[#References|[a2]]]. | (see [[#References|[a1]]], [[#References|[a2]]]). Generalized versions of these identities for curvature forms and curvature tensors of connections with torsion are given in [[#References|[a2]]]. | ||
Latest revision as of 09:36, 27 October 2014
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) |
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