Connections on bundles

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.

An Illustrated Handbook