The fibers of a smooth bundle \({(E,M,\pi)}\) let us define vertical tangents, but we have no structure that would allow us to canonically define a horizontal tangent. This structure is introduced via the **Ehresmann connection 1-form** (AKA bundle connection 1-form), a vector-valued 1-form on \({E}\) that defines the vertical component of its argument \({v}\), which we denote \({v^{⌽}}\), and therefore also defines the horizontal component, which we denote \({v^{⦵}}\):

\(\displaystyle \begin{aligned}\vec{\Gamma}(v) & \equiv v^{⌽},\\ H_{p} & \equiv\left\{ v\in T_{p}E\mid\vec{\Gamma}(v)=0\right\} \\ \Rightarrow v & =v^{⌽}+v^{⦵}, \end{aligned} \)

where \({v^{⌽}\in V_{p}}\), \({v^{⦵}\in H_{p}}\), and \({H_{p}}\) is called the **horizontal tangent space**. Viewing the \({H_{p}}\) as fibers over \({E}\) then yields the **horizontal bundle** \({(HE,E,\pi_{H})}\), and a **vertical form ** is defined to vanish whenever any of its arguments are horizontal. Alternatively, one can start by defining the horizontal tangent spaces as smooth sections of the jet bundle of order 1 over \({E}\), which uniquely determines a Ehresmann connection 1-form.

Δ “Ehresmann connection” can refer to the horizontal tangent spaces, the horizontal bundle, the connection 1-form, or the complementary 1-form that maps to the horizontal component of its argument. |

Recall that on a smooth principal bundle \({(P,M,\pi,G)}\), the right action \({\rho\colon G\rightarrow\mathrm{Diff}(P)}\) has a corresponding Lie algebra action \({\mathrm{d}\rho\colon\mathfrak{g}\rightarrow\mathrm{vect}(P)}\) where \({\mathrm{d}\rho\left|_{p}\right.}\) is a vector space isomorphism from \({\mathfrak{g}}\) to \({V_{p}}\). The **principal connection 1-form** (AKA principal \({G}\)-connection, \({G}\)-connection 1-form) is a \({\mathfrak{g}}\)-valued vertical 1-form \({\check{\Gamma}_{P}}\) on \({P}\) that defines the vertical part of its argument \({v}\) at \({p}\) via this isomorphism, i.e. the right action of the structure group transforms it into the Ehresmann connection 1-form:

\(\displaystyle \begin{aligned}\mathrm{d}\rho\left(\check{\Gamma}_{P}(v)\right)\left|_{p}\right. & \equiv v^{⌽}\\ & =\vec{\Gamma}(v) \end{aligned} \)

For \({g\in G}\), \({\mathrm{d}g(v)\colon T_{p}P\rightarrow T_{g(p)}P}\) is required to preserve horizontal tangent vectors as well as vertical. A principal connection 1-form exists on any principal bundle.

Δ As with the Ehresmann connection, a “connection” on a principal bundle can refer to the principal connection 1-form, the horizontal tangent spaces, or other related quantities. |