Connection on a principal bundle

Remarks

Equivalently, a connection is an Ehresmann connection $H \subset TP$ with $dR_g(H_u) = H_{u \cdot g}$ for all $u$ and $g$; the two notions correspond by $H = \ker \omega$. Example: on the trivial bundle $P = M \times G$, let $\theta$ be the left-invariant Maurer–Cartan form of $G$, so $\theta_g(v) = dL_{g^{-1}}(v)$, and let $\omega = \mathrm{pr}_2^* \theta$. Clause 1 holds since $X^\#_{(p, g)} = (0, dL_g X)$, and clause 2 holds since $R_h^* \theta = \mathrm{Ad}(h^{-1}) \theta$. Non-example: $\omega = 0$ satisfies clause 2 but fails clause 1 when $\dim G > 0$. For a matrix group the trivial example reads $\omega = g^{-1} dg$. The difference of two connection forms is a $\mathfrak{g}$-valued $1$-form that vanishes on vertical vectors and satisfies clause 2; so connections on $P$ form an affine space.