Principal connection

Let $\pi:P\to M$ be a principal G-bundle with structure Lie group $G$. For each $p\in P$, define the vertical subspace

A principal connection on $P$ is a smooth assignment of a subspace $H_p\subset T_pP$ (the horizontal subspace) for every $p\in P$ such that:

1. (Horizontal–vertical splitting) $T_pP = H_p \oplus V_p$ for all $p\in P$.
2. (Right-invariance) For every $g\in G$, the differential of the right action $R_g:P\to P$ satisfies $(R_g)_*(H_p)=H_{p\cdot g}.$

Equivalently, a principal connection is a $G$-equivariant splitting of the short exact sequence of vector bundles over $P$,

A principal connection can also be encoded by a connection 1-form $\omega$ on $P$; its kernel at each point is the horizontal subspace.

A principal connection determines horizontal lifts of curves and hence a notion of parallel transport in $P$.

Examples

1. Trivial bundle with a chosen gauge potential. On $P=U\times G\to U$, any $\mathfrak{g}$-valued 1-form $A\in \Omega^1(U;\mathfrak{g})$ defines a principal connection by declaring a tangent vector $(v,\xi)\in T_xU\oplus T_gG\cong T_{(x,g)}(U\times G)$ to be horizontal exactly when $\omega(v,\xi)=0$ for the standard connection form $\omega=\mathrm{Ad}(g^{-1})A + g^{-1}dg$. This makes the horizontal distribution explicit in a product trivialization.

2. Connections on frame bundles. If $M$ has a linear connection on its tangent bundle , then the frame bundle $\mathrm{Fr}(TM)\to M$ carries an induced principal connection whose horizontals correspond to parallel frames along curves.

3. Hopf fibration. The Hopf map $S^3\to S^2$ is a principal $U(1)$-bundle. The standard contact 1-form on $S^3$ defines a horizontal distribution (the orthogonal complement to the $U(1)$-orbits) that is invariant under the right $U(1)$-action, hence gives a principal connection.