Exterior covariant derivative

Fix a principal $G$-bundle $\pi:P\to M$ with a principal connection and associated connection form $\omega$. Let $V$ be a representation of $G$.

A $V$-valued $k$-form $\alpha\in \Omega^k(P;V)$ is called tensorial (of type $V$) if:

• (Horizontality) $\alpha(X_1,\dots,X_k)=0$ whenever one of the $X_i$ is vertical, and
• (Equivariance) $R_g^*\alpha=\rho(g^{-1})\alpha$, where $\rho:G\to \mathrm{GL}(V)$ is the representation.

Given such $\alpha$, the exterior covariant derivative is the operator

defined by

where $d$ is the exterior derivative and $X_i^H$ denotes the horizontal projection of $X_i\in T_pP$ using the connection (equivalently, the horizontal lift determined by $\ker\omega$).

This definition is independent of choices and produces another tensorial form. In local trivializations it becomes the familiar “$d$ plus connection term” formula, and its square is governed by the curvature: for tensorial $\alpha$,

where $\Omega$ is the curvature 2-form .

Examples

1. Equivariant functions (degree 0). If $f:P\to V$ is equivariant, then $d_\omega f$ is the horizontal part of $df$. Under the associated bundle viewpoint, this corresponds to the covariant derivative of the section defined by $f$.

2. Adjoint-valued local formula. For the adjoint representation $V=\mathfrak{g}$, the descended operator on local $\mathfrak{g}$-valued forms is $d_A=d+[A\wedge\cdot]$. This matches $d_\omega$ applied to tensorial forms on $P$.

3. Bianchi identity as a covariant closure. Applying $d_\omega$ to the curvature $\Omega$ yields $d_\omega \Omega=0$, the (first) Bianchi identity in principal-bundle form.