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The curvature of a principal connection is the horizontal equivariant two-formFor horizontal vector fields $X,Y$, it satisfies $\mathcal{F}(X,Y)=-\mathcal{A}([X,Y])$.

The horizontal distribution of a principal connection is the smooth complement $H$ to the tangent spaces of the group orbits. A tangent vector is horizontal exactly when the connection form $\mathcal{A}$ annihilates it.

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

LetThe vertical space is spanned by the fundamental vector $(y,-x)$ corresponding to $J\in \mathfrak{o}(2)$. DefineIt sends $(y,-x)$ to $J$, is $O(2)$-equivariant, and therefore is a principal connection. Its kernel consists exactly of those $(u,v)$ for whichThese are precisely the velocities satisfying the stated horizontality condition. Every tangent vector has the unique decompositioninto vertical and horizontal parts, proving uniqueness. This is the Canonical principal connection on the Stiefel bundle over a Grassmannian.

For a principal connection with connection form $\mathcal{A}$, the horizontal distribution of a principal connection is $H=\ker \mathcal{A}$, the complement of the tangent spaces to the $G$-orbits. Its curvature of a principal connection isIf $X,Y$ are horizontal vector fields, then $\mathcal{A}(X)=\mathcal{A}(Y)=0$, and henceThe Frobenius theorem says that $H$ is integrable exactly when $[X,Y]$ is horizontal for all horizontal $X,Y$. The displayed identity makes this equivalent to the vanishing of the horizontal two-form $\mathcal{F}$, hence to $\mathcal{F}=0$. Thus the horizontal distribution is integrable exactly for a flat principal connection.