\( \newcommand{\bs}{\boldsymbol}
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\partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}}
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\newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)
X. Curvature
The second Bianchi identity is a differential identity
obtained by taking the exterior coderivative of the curvature
2-form:
\[{\text{D}}{{\mathbf{R}}^a}_b
= {\text{d}}{{\mathbf{R}}^a}_b + {{\bs{\omega }}^a}_c \wedge
{{\mathbf{R}}^c}_b - {{\bs{\omega }}^c}_b \wedge
{{\mathbf{R}}^a}_c = 0\]
Proof: Substitute the second Cartan equation (10.6) and use
the property ${\text{dd}}{\mathbf{A}} = 0$ for general
${\mathbf{A}}$; see definition (1.7). After some algebra, taking
into account the (anti-)commutation rule ${( - 1)^p}$, one obtains
(10.13). [Wikipedia:
Spin Connection]
For the case of no torsion, the second Bianchi identity may
be written in the component form
\[{D_\rho }{R_{\mu \nu \kappa
\lambda }} + {D_\mu }{R_{\nu \rho \kappa \lambda }} + {D_\nu
}{R_{\rho \mu \kappa \lambda }} = 3{D_{[\rho }}{R_{\mu \nu ]\kappa
\lambda }} = 0\]
This equation may be also proven on the basis of the Jacobi
identity, since it basically expresses
\[[[{D_{[\mu }},{D_\nu
}],{D_{\kappa ]}}] = 0\]
By operating with the Jacobi operator on an arbitrary vector and
using definitions (10.1) and (10.8), one derives the second Bianchi
identity in the cyclic form
\[{D_\rho }\operatorname{Riem}
({{\mathbf{g}}_\mu } \wedge {{\mathbf{g}}_\nu }) + {\text{cyclic}} =
0\]
The above version of the second Bianchi identity is true if and
only if the connection is symmetric and the torsion is
zero.