*4.1. For an Ehresmann Connection HE*

**Observation 6:** Ω*<sup>k</sup> G* (*P*, *V*) is not closed under the ordinary exterior derivative. In that sense, if *<sup>α</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>k</sup> G* (*P*, *<sup>V</sup>*), then *<sup>d</sup><sup>α</sup>* <sup>∈</sup>/ <sup>Ω</sup>*k*+<sup>1</sup> *G* (*P*, *V*). This is what a covariant differentiation will do instead.

The idea of a covariant exterior derivative for a connection *HE* is, given such an Ehresmann connection *HE*, the one of projecting vector fields onto this horizontal bundle and then feed our ordinary exterior derivative with such horizontal vector fields.

First of all, we define a map acting as a pull-back. Namely that, given a map *h* : *TP* → *HE* such that, for all vertical vector fields *v*, we get *h* ◦ *v* := *hv* = 0 (called the *horizontal projection*), we define the dual map *h* ∗ : *T* <sup>∗</sup>*P* → *HE*<sup>∗</sup> such that, for *<sup>α</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> (*P*, *V*) and *V* a vector space, we have *h* <sup>∗</sup> ◦ *α* := *h* <sup>∗</sup>*α* = *α* ◦ *h*.

**Definition 15** (*d h* )**.** *Let <sup>P</sup> be a G-principal bundle, <sup>V</sup> be a vector space, and <sup>α</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>k</sup>* (*P*, *V*) *be an equivariant form. We define the exterior covariant derivative d<sup>h</sup> as a map d<sup>h</sup>* : Ω*<sup>k</sup>* (*P*, *<sup>V</sup>*) <sup>→</sup> <sup>Ω</sup>*k*+<sup>1</sup> *G* (*P*, *V*) *such that*

$$d^!a(v\_0, \ldots, v\_k) := h^\* d a(v\_0, \ldots, v\_k) = d a(hv\_0, \ldots, hv\_k),\tag{26}$$

*where v*0, ..., *v<sup>k</sup> are vector fields.*

It depends on the choice of our Ehresmann connection *HE*, which reflects onto the horizontal projection *h*; that is why we have the index *<sup>h</sup>* .

**Observation 7:** We can make our covariant derivative depend only on *ω*, if we restrict it to only forms in Ω*<sup>k</sup> G* (*P*, *V*) and if we consider the representation of the algebra induced by the derivative of *ρ* that we denote *<sup>d</sup><sup>ρ</sup>* : <sup>g</sup> <sup>→</sup> End(*V*). Then, we have *<sup>d</sup><sup>ρ</sup>* ◦ *<sup>ω</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>k</sup> P*, End(*V*) .
