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 an \({\mathbb{K}^{n}}\)-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.

An Illustrated Handbook