A principal bundle (AKA principal \({G}\)-bundle) \({(P,M,\pi,G)}\) has a topological group \({G}\) as both abstract fiber and structure group, where \({G}\) acts on itself via left translation as a transition function across trivializing neighborhoods, i.e.
\(\displaystyle f_{i}(p)=g_{ij}f_{j}(p), \)
where the operation of \({g_{ij}}\) is the group operation. Note that the fiber over a point \({\pi^{-1}(x)}\) is only homeomorphic as a space to \({G}\) in a given trivializing neighborhood, and so is missing a unique identity element and is a \({G}\)-torsor, not a group (see the section on group actions).
A principal bundle lets us introduce a consistent right action of \({G}\) on \({\pi^{-1}(x)}\) (as opposed to the left action on the abstract fiber). This right action is defined by
\(\displaystyle \begin{aligned}g(p) & \equiv f_{i}^{-1}\left(f_{i}(p)g\right)\\ \Rightarrow f_{i}\left(g(p)\right) & =f_{i}(p)g \end{aligned} \)
for \({p\in\pi^{-1}(U_{i})}\), where in an intersection of trivializing neighborhoods \({U_{i}\cap U_{j}}\) we see that
\(\displaystyle \begin{aligned}g(p) & =f_{j}^{-1}\left(f_{j}(p)g\right)\\ & =f_{i}^{-1}f_{i}f_{j}^{-1}\left(f_{j}(p)g\right)=f_{i}^{-1}\left(g_{ij}f_{j}(p)g\right)\\ & =f_{i}^{-1}\left(f_{i}(p)g\right)=g(p), \end{aligned} \)
i.e. \({g(p)}\) is consistently defined across trivializing neighborhoods. Via this fiber-wise action, \({G}\) then has a right action on the bundle \({P}\).
The above depicts how a principal bundle has the same group \({G}\) as both abstract fiber and structure group, where \({G}\) acts on itself via left translation. \({G}\) also has a right action on the bundle itself, which is consistent across trivializing neighborhoods. The identity sections (defined below) are also depicted.
| Δ It is important to remember that \({M}\) is not part of \({E}\), and that the depiction of each fiber in the bundle \({\pi^{-1}(x)\in E}\) as “hovering over” the point \({x\in M}\) is only valid locally. |
| Δ Note that from its definition and basic group properties, the right action of \({G}\) on \({\pi^{-1}(x)}\) is automatically free and transitive (making \({\pi^{-1}(x)}\) a “right \({G}\)-torsor”). An equivalent definition of a principal bundle excludes \({G}\) as a structure group but includes this free and transitive right action of \({G}\). Also note that the definition of the right action is equivalent to saying that \({f_{i}\colon\pi^{-1}(x)\rightarrow G}\) is equivariant with respect to the right action of \({G}\) on \({\pi^{-1}(x)}\) and the right action of \({G}\) on itself. |
| Δ A principal bundle is sometimes defined so that the structure group acts on itself by right translation instead of left. In this case the action of \({G}\) on the bundle must be a left action. |
| Δ A principal bundle can also be denoted \({P(M,G)}\) or \({G\hookrightarrow P\overset{\pi}{\longrightarrow}M}\). |
Since the right action is an intrinsic operation, a principal bundle map between principal \({G}\)-bundles (e.g. a principal bundle automorphism) is required to be equivariant with regard to it, i.e. we require \({\Phi_{E}(g(p))=g(\Phi_{E}(p))}\), or in juxtaposition notation, \({\Phi_{E}(pg)=\Phi_{E}(p)g}\). In fact, any such equivariant map is automatically a principal bundle map, and if the base spaces are identical and unchanged by \({\Phi_{E}}\), then \({\Phi_{E}}\) is an isomorphism. For a principal bundle map \({\Phi_{E}\colon(P^{\prime},M^{\prime},G^{\prime})\rightarrow(P,M,G)}\) between bundles with different structure groups, we must include a homomorphism \({\Phi_{G}\colon G^{\prime}\rightarrow G}\) between structure groups so that the equivariance condition becomes \({\Phi_{E}(g(p))=\Phi_{G}(g)(\Phi_{E}(p))}\), or in juxtaposition notation, \({\Phi_{E}(pg)=\Phi_{E}(p)\Phi_{G}(g)}\).
| Δ Note that the right action of a fixed \({g\in G}\) is thus not a principal bundle automorphism, since for non-abelian \({G}\) it will not commute with another right action. |
A principal bundle has a global section iff it is trivial. However, within each trivializing neighborhood on a principal bundle we can define a local identity section
\(\displaystyle \sigma_{i}(x)\equiv f_{i}^{-1}(e), \)
where \({e}\) is the identity element in \({G}\). In \({U_{i}\cap U_{j}}\), we can then use \({f_{i}(\sigma_{i})=e}\) to see that the identity sections are related by the right action of the transition function:
\(\displaystyle \begin{aligned}g_{ij}(\sigma_{i}) & =f_{i}^{-1}\left(f_{i}(\sigma_{i})g_{ij}\right)\\ & =f_{i}^{-1}\left(g_{ij}\right)\\ & =f_{i}^{-1}\left(g_{ij}f_{j}(\sigma_{j})\right)\\ & =f_{i}^{-1}\left(f_{i}(\sigma_{j})\right)\\ & =\sigma_{j}, \end{aligned} \)
or in juxtaposition notation,
\(\displaystyle \sigma_{j}=\sigma_{i}g_{ij}. \)
| Δ The different actions of \({G}\) are a potential source of confusion. \({g_{ij}}\) has a left action on the abstract fiber of a \({G}\)-bundle, which on a principal bundle becomes left group multiplication, and also has a right action on the bundle itself that relates the elements in the identity section. |
If \({G}\) is a closed subgroup of a Lie group \({P}\) (and thus also a Lie group by Cartan’s theorem), then \({(P,P/G,G)}\) is a principal \({G}\)-bundle with base space the (left) coset space \({P/G}\). The right action of \({G}\) on the entire space \({P}\) is just right translation.