**Definition 9** (Vertical bundle)**.** *Let M be a differentiable manifold and π* : *E* → *M be a smooth fiber bundle. We call the sub-bundle VE* = Ker(*π*<sup>∗</sup> : *TE* → *TM*) *the vertical bundle.*

Following this definition, we have the natural extension to the complementary bundle of the vertical bundle, which is somehow the formalization of the idea we had of a bundle that takes care of tangent vector fields on *M*.

**Definition 10** (Ehresmann connection)**.** *Let M be a differentiable manifold and π* : *E* → *M be a smooth fiber bundle. Consider a complementary bundle HE such that TE* = *HE* ⊕ *VE. We call this smooth sub-bundle HE the horizontal bundle or Ehresmann connection.*

Thus, vector fields will be called *vertical* or *horizontal* depending on whether they belong to Γ(*VE*) or Γ(*HE*), respectively.
