Covariant exterior derivative preserves tensoriality

Let $\pi:P\to M$ be a principal G-bundle with right action $R_g:P\to P$, and let $\rho:G\to \mathrm{GL}(V)$ be a finite-dimensional representation with differential $\rho_*:\mathfrak g\to \mathfrak{gl}(V)$ (see representation of a Lie group ).

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

1. (Horizontality) $\alpha_p(v_1,\dots,v_k)=0$ whenever at least one $v_i$ is vertical (i.e. tangent to the fiber), equivalently $\iota_{X^\#}\alpha=0$ for all fundamental vertical vector fields $X^\#$ (compare vertical subbundle and fundamental vector fields ).
2. ($\rho$-equivariance) $(R_g)^*\alpha=\rho(g^{-1})\alpha$ for all $g\in G$ (compare equivariance ).

Fix a principal connection with connection $1$-form $\omega\in \Omega^1(P;\mathfrak g)$ (see connection 1-form ). Let $\mathrm{hor}:TP\to TP$ denote the horizontal projection determined by $\omega$. The covariant exterior derivative of a $V$-valued $k$-form is the operator

Equivalently, on tensorial forms one may write the usual local formula

where the wedge combines $\mathfrak{gl}(V)$ acting on $V$ with the exterior product.

Lemma

If $\alpha$ is tensorial of type $\rho$, then $D^\omega\alpha$ is also tensorial of type $\rho$.

In other words, the covariant exterior derivative restricts to a well-defined map

where $\Omega^k_{\mathrm{tens}}(P;V)$ denotes $V$-valued tensorial $k$-forms. This is the basic mechanism behind induced connections on associated bundles and, in the vector bundle case, on associated vector bundles .

Examples

1. From equivariant functions to covariant derivatives.
   A tensorial $0$-form of type $\rho$ is just a $\rho$-equivariant function $f:P\to V$. Under the identification of such functions with sections of the associated bundle $E=P\times_\rho V$ (see equivariant maps and sections ), the tensorial $1$-form $D^\omega f$ corresponds to the covariant derivative $\nabla s$ of the associated section $s\in \Gamma(E)$ (see covariant derivative of a section ).

2. Adjoint-valued forms and the Bianchi identity setting.
   Taking $V=\mathfrak g$ with the adjoint representation (see adjoint action ), tensorial $\mathfrak g$-valued forms are the same objects that appear in covariant exterior derivatives on adjoint-valued forms . In particular, the curvature form $F^\omega\in\Omega^2(P;\mathfrak g)$ (see curvature 2-form ) is tensorial, hence $D^\omega F^\omega$ is a well-defined tensorial $3$-form.

3. Frame bundle viewpoint.
   On the frame bundle of a vector bundle (see frame bundle construction ), tensorial forms of the defining representation encode tensor fields on the base. The lemma guarantees that applying $D^\omega$ to such a tensorial form produces another tensorial form, which is exactly what is needed for defining covariant derivatives of tensor fields via the frame bundle picture (compare connections via frame bundles ).