Principal connection 1-form from a horizontal distribution

Write $p=s(x)a$ in an equivariant principal chart, with $s$ its local section and $a\in G$. Then ${\varphi }_p(h)=ph=s(x)(ah)$; freeness and transitivity give bijectivity on the fibre, and this coordinate formula gives a smooth inverse. Hence ${\varphi }_p:G\to P_x$, $h\mapsto ph$, is a diffeomorphism, so its differential at $e$ is a linear isomorphism $T_{e}G\to T_p{P}_x=V_{p}P$. For each $t\in \mathbb{R}$, the curve $c_t(s):={\gamma }_Z(ts)$ is a one-parameter subgroup with initial tangent $tZ$; uniqueness in [F4] gives $c_t={\gamma }_{tZ}$. Thus ${\exp }_G(tZ)={\gamma }_{tZ}(1)={\gamma }_Z(t)$. The curve $t\mapsto p{\exp }_G(tZ)$ is therefore ${\varphi }_p\circ {\gamma }_Z$, whose derivative at zero is $d({\varphi }_p{)}_{e}Z$; hence ${\zeta }_p$ is the stated isomorphism.

In the same chart, ${\zeta }_p$ is the derivative of the smooth action map and its inverse is the derivative of the inverse fibre coordinate, so the family $\zeta$ and its inverse are smooth bundle maps. The complement $TP=VP\oplus H$ gives a smooth bundle projection $v$, so $\omega ={\zeta }^{-1}\circ v$ is a smooth fibrewise-linear 1-form. On a vertical vector $Z_P(p)={\zeta }_p(Z)$, the projection is the vector itself; therefore $\omega (Z_P)=Z$. Also $v_{p}X=0$ exactly when $X\in H_p$, so $\ker \omega =H$.

For $g\in G$, let $R_g(p)=pg$. The maps $R_g\circ {\varphi }_p$ and ${\varphi }_{pg}\circ C_{g^{-1}}$ agree because $phg=pg(g^{-1}hg)$. Differentiating at $e$ by [F6] gives $d(R_g{)}_p{Z}_P(p)=({\mathrm{Ad}}_{g^{-1}}Z{)}_P(pg)$. Since $H$ and $VP$ are right-invariant, their projections commute with $d{R}_g$; applying the definition of $\omega$ yields ${\omega }_{pg}(d{R}_{g}X)={\mathrm{Ad}}_{g^{-1}}{\omega }_p(X)$.

Conversely, if $\omega$ has the stated reproduction and equivariance properties, define $v:=\zeta \circ \omega$ and $H:=\ker \omega$. Reproduction gives $v\circ \zeta =\zeta$, hence $v^2=v$, $\mathrm{im}v=VP$, and $TP=VP\oplus H$. Equivariance of $\omega$ and the orbit differential identity from step 2.2 make $H$ right-invariant; smoothness of $\omega$ makes it a smooth subbundle. Thus this is a principal connection and the construction is inverse to step 2.1.

The imported curvature $K_H$ is a smooth alternating map with values in $VP$, and $\omega$ restricts to the inverse of $\zeta :P\times \mathfrak{g}\to VP$; consequently ${\kappa }_H=\omega \circ K_H$ is a well-defined smooth $\mathfrak{g}$-valued alternating 2-form. This is only the conversion of its target fibre; the Cartan identity is proved separately.