View source for Bianchi identity

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:

<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>

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bianchi,   "Lezioni di geometria differenziale" , '''1–2''' , Zanichelli , Bologna  (1923–1927)</TD></TR></table>



====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.

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>

(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]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.J. Hicks,   "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>