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 finite-dimensional real or complex smooth 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

d_\omega:\Omega^k_{\mathrm{tens}}(P;V)\to \Omega^{k+1}_{\mathrm{tens}}(P;V)

defined by

(d_\omega \alpha)_p(X_0,\dots,X_k)\coloneqq (d\alpha)_p(X_0^H,\dots,X_k^H),

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 $\ker\omega$ ).

Tensoriality and curvature

This definition produces another tensorial form. In a local trivialization it becomes the “ $d$ plus connection term” formula. Its square is governed by the curvature:

d_\omega^2\alpha=\rho_*(\Omega)\wedge\alpha,

where $\Omega$ is the curvature form and $\rho_*:\mathfrak g\to\operatorname{End}(V)$ is the derived representation.

Examples

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 $f$ .

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 $P$ .

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.