A relation connecting the components of the covariant derivatives of the curvature tensor of a Riemannian space:
where . First established by L. Bianchi  in 1902.
|||L. Bianchi, "Lezioni di geometria differenziale" , 1–2 , Zanichelli , Bologna (1923–1927)|
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
|[a1]||N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)|
|[a2]||S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)|
Bianchi identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bianchi_identity&oldid=13332