 Solution

A principal connection on $P$ is a $G$-equivariant smooth splittingwhere $V_p$ is tangent to the $G$-orbit. Equivalently, it is a $\mathfrak{g}$-valued one-form $\mathcal{A}$ satisfying $\mathcal{A}({\xi }_P)=\xi$ and $R_g^{\ast }\mathcal{A}={\mathrm{Ad}}_{g^{-1}}\mathcal{A}$. Its curvature of a principal connection is