Let $\nu: P \to M$ be a smooth principal right $G$-bundle over a smooth manifold $M$, and let $\mathfrak g = T_eG$ be the Lie algebra of $G$. For $\xi \in \mathfrak g$, let $\xi_P \in \mathfrak X(P)$ denote the fundamental vector field
paragraph
admin
\begin{align*}
(\xi_P)_p = \left.\frac{d}{dt}\right|_{t=0} p \cdot \exp(t\xi).
\end{align*}
latex_env
admin
There is a bijection between principal connections, that is, smooth subbundles $HP \subset TP$ satisfying
for every $p \in P$ and $g \in G$, and connection forms, that is, smooth $\mathfrak g$-valued $1$-forms $\omega \in \Omega^1(P;\mathfrak g)$ satisfying
for every $g \in G$. The bijection sends a principal connection $HP$ to the form that records the vertical component of a tangent vector under the identification $\mathfrak g \cong \ker d\nu_p$, and sends a connection form $\omega$ to the horizontal distribution $\ker \omega$.