Principal G-bundle - Knowlpedia

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

P\times G\to P,\quad (p,g)\mapsto p\cdot g,

such that:

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.

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).

Underlying bundle and structure group

Forgetting the right action leaves a smooth fiber bundle with model fiber $G$ . In equivariant local trivializations, changes of fiber coordinates have the form $(x,h)\mapsto(x,g_{ij}(x)h)$ . Thus this is a G-bundle for the left multiplication action of $G$ on itself. That left action in the transition functions commutes with the principal right action on $P$ .

Examples

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.

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

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.