Bianchi identity

The covariant exterior derivative of the curvature form of a connection vanishes.

Bianchi identity

Let $\pi:P\to M$ be a principal $G$ -bundle, and let $\omega \in \Omega^1(P;\mathfrak{g})$ be a connection $1$ -form. The curvature is the $\mathfrak{g}$ -valued $2$ -form

\Omega \;=\; d\omega \;+\; \frac{1}{2}[\omega \wedge \omega],

where $[\omega\wedge\omega]$ denotes the wedge product combined with the Lie bracket on $\mathfrak{g}$ .

Define the covariant exterior derivative $D_\omega$ acting on $\mathfrak{g}$ -valued forms $\eta \in \Omega^k(P;\mathfrak{g})$ by

D_\omega \eta \;=\; d\eta \;+\; [\omega \wedge \eta].

Identity (second/Bianchi identity)

The curvature satisfies

D_\omega \Omega = 0.

Interpretation and consequences

$D_\omega \Omega=0$ is the bundle/form version of the “second Bianchi identity.” In tensor notation for a covariant derivative $\nabla$ on the base, it corresponds to the cyclic identity $(\nabla_X R)(Y,Z) + (\nabla_Y R)(Z,X) + (\nabla_Z R)(X,Y) = 0,$ where $R$ is the curvature tensor.
In Chern–Weil theory, the Bianchi identity implies that characteristic forms built from $\Omega$ are closed. Combined with the basic forms theorem , this explains how these forms descend from $P$ to well-defined de Rham cohomology classes on $M$ .