Covariant exterior derivative on ad(P)-valued forms

Let $\pi:P\to M$ be a principal G-bundle with Lie algebra $\mathfrak{g}$. The adjoint bundle is the vector bundle

associated to the adjoint action of $G$ on the Lie algebra $\mathfrak{g}$.

A principal connection on $P$ induces a connection $\nabla$ on $\mathrm{ad}(P)$ (equivalently, it induces the operator on tensorial forms described by the exterior covariant derivative ). The covariant exterior derivative

is the unique graded derivation extending $\nabla$ and satisfying the usual Koszul formula: for $\alpha\in \Omega^k(M;\mathrm{ad}(P))$ and vector fields $X_0,\dots,X_k$,

where the bracket is the Lie bracket of vector fields on $M$.

Locally, if $A\in \Omega^1(U;\mathfrak{g})$ is a local connection form and we identify $\mathrm{ad}(P)|_U\cong U\times \mathfrak{g}$, then for $\alpha\in \Omega^k(U;\mathfrak{g})$,

with $d$ the exterior derivative .

Examples

1. Sections of $\mathrm{ad}(P)$ (degree 0). For $\phi\in \Omega^0(M;\mathrm{ad}(P))$, locally $\phi:U\to\mathfrak{g}$, one has

   $$d_\nabla \phi = d\phi + [A,\phi],$$

   which is the usual covariant derivative in the adjoint representation.

2. Bianchi identity on the base. Let $F\in \Omega^2(U;\mathfrak{g})$ be the local curvature 2-form. Then

   $$d_\nabla F = 0$$

   is the Bianchi identity written as a covariant closure condition.

3. Abelian structure group. If $G$ is abelian, then $[A\wedge \alpha]=0$ for all $\alpha$, so $d_\nabla$ reduces to the ordinary exterior derivative on $\mathfrak{g}$-valued forms.