Principal connection

A principal connection can be specified by a $G$-equivariant horizontal distribution $H\subset TP$ complementary to the vertical bundle, or equivalently by a Lie-algebra-valued connection form $\mathcal{A}$ that reproduces infinitesimal generators and has $H=\ker \mathcal{A}$.

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 local section $s:U\to P$ is horizontal when $ds(TU)\subseteq H$, equivalently when $s^{\ast }\mathcal{A}=0$. A flat principal connection has horizontal sections locally, while its holonomy can obstruct a global horizontal section.

The holonomy of a connection along a closed curve is the group element relating the endpoints of its horizontal lift. A flat connection can have nontrivial holonomy around a noncontractible loop.

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

A principal connection is flat when its curvature vanishes. The identity $\mathcal{F}(X,Y)=-\mathcal{A}([X,Y])$ shows that this is equivalent to integrability of its horizontal distribution.

For the principal $O(k)$-bundle of orthonormal $k$-frames over the real Grassmannian, the canonical horizontal vectors move every frame vector orthogonally to the spanned plane. If a frame is represented by a matrix $Q$ with $Q^{\mathsf{T}}Q=I$, the connection form is