Principal G-bundle

Let $G$ be a Lie group and let $M$ be a smooth manifold. A principal $G$-bundle over $M$ is a quadruple $(P,\pi,M,G)$ consisting of a smooth manifold $P$, a surjective submersion $\pi:P\to M$, and a smooth right action

such that:

1. Free and transitive on fibers. For each $x\in M$, the action restricts to a free and transitive action of $G$ on the fiber $P_x:=\pi^{-1}(x)$. Equivalently, each fiber is a $G$-torsor.

2. Local triviality (equivariant). There exists an open cover $\{U_\alpha\}$ of $M$ and diffeomorphisms

   $$\Phi_\alpha:\pi^{-1}(U_\alpha)\to U_\alpha\times G$$

   such that:

   • $\mathrm{pr}_1\circ \Phi_\alpha=\pi$,
   • $\Phi_\alpha(p\cdot g) = \Phi_\alpha(p)\cdot g$, where $U_\alpha\times G$ has the right action $(x,h)\cdot g := (x,hg)$.

A morphism of principal $G$-bundles over the same base is a smooth $G$-equivariant map $\Psi:P\to P'$ commuting with projections to $M$.

Principal bundles are the natural setting for principal connections ; once a connection is chosen, one can define parallel transport along paths and the associated holonomy group , and the curvature measures the failure of horizontal distributions to be integrable (see curvature ).

Examples

1. Trivial principal bundle. For any $M$ and $G$, the projection $M\times G\to M$ with right action $(x,h)\cdot g=(x,hg)$ is a principal $G$-bundle.

2. Frame bundles. If $E\to M$ is a rank-$n$ vector bundle, its frame bundle is a principal bundle with structure group $\mathrm{GL}(n,\mathbb F)$.

3. Hopf fibration. The map $S^3\to S^2$ can be realized as a principal $\mathrm{U}(1)$-bundle, with $\mathrm{U}(1)$ acting freely on $S^3\subset \mathbb C^2$ by scalar multiplication.