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Difference between revisions of "Bianchi identity"

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A relation connecting the components of the covariant derivatives of the [[Curvature tensor|curvature tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016020/b0160201.png" /> of a Riemannian space:
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A relation connecting the components of the covariant derivatives of the [[Curvature tensor|curvature tensor]] $R_{ijk}^h$ of a Riemannian space:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016020/b0160202.png" /></td> </tr></table>
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$$R_{ijk,l}^h+R_{ikl,j}^h+R_{ilj,k}^h=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016020/b0160203.png" />. First established by L. Bianchi [[#References|[1]]] in 1902.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016020/b0160204.png" /> denotes of course the covariant derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016020/b0160205.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016020/b0160206.png" />-th coordinate.
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016020/b0160207.png" /></td> </tr></table>
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$$R_{jkl}^i+R_{klj}^i+R_{ljk}^i=0$$
  
 
(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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article