# The exterior covariant derivative on bundles

The exterior covariant derivative of a form on a smooth bundle with connection is the horizontal form that results from taking the exterior derivative on the horizontal components of all its arguments, i.e. for a $${k}$$-form $${\varphi}$$ we define

\displaystyle \begin{aligned}\mathrm{D}\varphi(v_{0},\ldots,v_{k}) & \equiv\mathrm{d}\varphi(v_{0}^{⦵},\ldots,v_{k}^{⦵}).\end{aligned}

On a smooth bundle, $${\mathrm{D}\varphi}$$ can then be viewed as the “sum of $${\varphi}$$ on the boundary of the horizontal hypersurface defined by its arguments.” Note that these boundaries are all defined by horizontal vectors except those including a Lie bracket, which may have a vertical component. So for example, if $${\varphi}$$ is a vertical 1-form we have $${\mathrm{D}\varphi(v,w)=-\varphi(\left[v^{⦵},w^{⦵}\right])}$$, the other terms all vanishing.

For a vector bundle $${(E,M,\mathbb{K}^{n})}$$ associated to a smooth principal bundle with connection $${(P,M,GL(n,\mathbb{K}))}$$, it can be shown that a $${\mathbb{K}^{n}}$$-valued horizontal equivariant form $${\vec{\varphi}_{P}}$$ on $${P}$$ satisfies the familiar equation

$$\displaystyle \mathrm{D}\vec{\varphi}_{P}=\mathrm{d}\vec{\varphi}_{P}+\check{\Gamma}_{P}\wedge\vec{\varphi}_{P},$$

where as usual the derivatives are taken on the components of $${\vec{\varphi}_{P}}$$, and the action of $${gl(n,\mathbb{K})}$$-valued $${\check{\Gamma}_{P}}$$ on the values of $${\vec{\varphi}_{P}}$$ is the differential of the left action of $${GL(n,\mathbb{K})}$$. $${\mathrm{D}\vec{\varphi}_{P}}$$ is then also a horizontal equivariant form. Applying the pullback by the identity section to the exterior covariant derivative, we obtain the expected

$$\displaystyle \mathrm{D}\vec{\varphi}_{i}=\mathrm{d}\vec{\varphi}_{i}+\check{\Gamma}_{i}\wedge\vec{\varphi}_{i}.$$

 Δ As with the connection 1-form, it is important to remember that the values of $${\vec{\varphi}_{i}}$$ on $${M}$$ are components operated on by the matrix $${\check{\Gamma}_{i}}$$, both of which are defined by a local trivialization.

The immediate application of the above is to a $${\mathbb{K}^{n}}$$-valued form on the frame bundle. However, we can also apply it to other associated bundles to $${P}$$. In particular, recalling the section on vertical tangents, in the associated bundle $${(\mathrm{Ad}P,M,gl(n,\mathbb{K}))}$$ we can apply it to a $${gl(n,\mathbb{K})}$$-valued horizontal equivariant form $${\check{\Theta}_{P}}$$ on $${P}$$, where the left action of $${GL(n,\mathbb{K})}$$ is $${\rho=\mathrm{Ad}}$$, and the left action of $${gl(n,\mathbb{K})}$$ on itself is therefore $${\mathrm{d}\rho=\mathrm{ad}}$$, i.e. the Lie bracket. For such a form we then have

$$\displaystyle \mathrm{D}\check{\Theta}_{P}=\mathrm{d}\check{\Theta}_{P}+\check{\Gamma}_{P}[\wedge]\check{\Theta}_{P},$$

where again the exterior derivative is taken on the matrix components of $${\check{\Theta}_{P}}$$, and the action of $${gl(n,\mathbb{K})}$$-valued $${\check{\Gamma}_{P}}$$ on the values of $${\check{\Theta}_{P}}$$ is the Lie bracket. Applying the pullback by the identity section recovers the same formula for algebra-valued forms on $${M}$$, as previously seen.