Editing Principal bundle (section)

==Basic properties==

===Trivializations and cross sections===

One of the most important questions regarding any fiber bundle is whether or not it is [[trivial bundle|trivial]], ''i.e.'' isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:

:'''Proposition'''. ''A principal bundle is trivial if and only if it admits a global [[Section (fiber bundle)|section]].''

The same is not true in general for other fiber bundles. For instance, [[vector bundle]]s always have a zero section whether they are trivial or not and [[fiber bundle#Sphere bundles|sphere bundles]] may admit many global sections without being trivial.

The same fact applies to local trivializations of principal bundles. Let {{math|''π'' : ''P'' → ''X''}} be a principal {{math|''G''}}-bundle. An [[open set]] {{math|''U''}} in {{math|''X''}} admits a local trivialization if and only if there exists a local section on {{math|''U''}}. Given a local trivialization

:<math>\Phi : \pi^{-1}(U) \to U \times G</math>
one can define an associated local section 
:<math>s:U \to \pi^{-1}(U);s(x) = \Phi^{-1}(x,e)\,</math>
where {{math|''e''}} is the [[identity element|identity]] in {{math|''G''}}. Conversely, given a section {{math|''s''}} one defines a trivialization {{math|Φ}} by
:<math>\Phi^{-1}(x,g) = s(x)\cdot g.</math>
The simple transitivity of the {{math|''G''}} action  on the fibers of {{math|''P''}} guarantees that this map is a [[bijection]], it is also a [[homeomorphism]]. The local trivializations defined by local sections are {{math|''G''}}-[[equivariant]] in the following sense. If we write
:<math>\Phi : \pi^{-1}(U) \to U \times G</math>
in the form 
:<math>\Phi(p) = (\pi(p), \varphi(p)),</math>
then the map
:<math>\varphi : P \to G</math>
satisfies
:<math>\varphi(p\cdot g) = \varphi(p)g.</math>
Equivariant trivializations therefore preserve the {{math|''G''}}-torsor structure of the fibers. In terms of the associated local section {{math|''s''}} the map {{math|''φ''}} is given by
:<math>\varphi(s(x)\cdot g) = g.</math>
The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

Given an equivariant local trivialization {{math|({''U''<sub>''i''</sub>}, {Φ<sub>''i''</sub>})}} of {{math|''P''}}, we have local sections {{math|''s''<sub>''i''</sub>}} on each {{math|''U''<sub>''i''</sub>}}. On overlaps these must be related by the action of the structure group {{math|''G''}}. In fact, the relationship is provided by the [[Transition map|transition functions]]
:<math>t_{ij} : U_i \cap U_j \to G\,.</math>
By gluing the local trivializations together using these transition functions, one may reconstruct the original principal bundle. This is an example of the [[fiber bundle construction theorem]].
For any {{math|''x'' ∈ ''U''<sub>''i''</sub> ∩ ''U''<sub>''j''</sub>}} we have
:<math>s_j(x) = s_i(x)\cdot t_{ij}(x).</math>

===Characterization of smooth principal bundles===

If <math>\pi : P \to X</math> is a smooth principal <math>G</math>-bundle then <math>G</math> acts freely and [[proper map|properly]] on <math>P</math> so that the orbit space <math>P/G</math> is [[diffeomorphic]] to the base space <math>X</math>. It turns out that these properties completely characterize smooth principal bundles. That is, if <math>P</math> is a smooth manifold, <math>G</math> a Lie group and <math>\mu : P \times G \to P</math> a smooth, free, and proper right action then
*<math>P/G</math> is a smooth manifold,
*the natural projection <math>\pi : P \to P/G</math> is a smooth [[submersion (mathematics)|submersion]], and
*<math>P</math> is a smooth principal <math>G</math>-bundle over <math>P/G</math>.