Principal bundles

First approach

Definition and remarks

We come from the context G-bundles.
Given a G-bundle $P \overset{π}{\to} M$ we will call it a principal $G$ -bundle if the standard fibre is $S = G$ and there exists a free right action $ρ$ of $G$ on $P$ such that

The orbits of $ρ$ are the fibres of $P$ .
For every local trivialization

ψ : U_{} \times G ⟶ π^{- 1} (U)

we have that for every $m \in U$ and $h, g \in G$

\tilde{ψ} (m) (h g) = ρ (\tilde{ψ} (m) (h), g)

Observe that this is equivalent to saying that we have a fibrewise right action of $G$ over $P$ that is free and transitive over each fibre. Each fibre is, then, a G-torsor. We will use indistinctly the notations $ρ_{g} (p) = ρ (p, g) = p g = R_{g} (p)$ when needed for clarity.

We will write
Pasted image 20211231171000.png|400

Infinitesimal behavior of the right action

Let $P$ be a right principal $G$ -bundle over a smooth manifold $M$ , and let $π : P \to M$ be the projection map. Let $R_{g} : P \to P$ denote the right action of the group $G$ on $P$ , defined by $R_{g} (p) = p \cdot g$ for $p \in P$ and $g \in G$ .

We consider the differential of the projection map, denoted $d π : T P \to T M$ , which is a linear map between the tangent spaces at corresponding points. Moreover, let $(R_{g})_{*} : T P \to T P$ be the pushforward (differential) of $R_{g}$ .

The behavior of $d π$ with respect to $(R_{g})_{*}$ is characterized by the following commutative diagram:

\begin{array}{ccc} T P & \overset{(R_{g})_{*}}{\to} & T P \\ ↓ d π & ↓ d π \\ T M & = & T M \end{array}

This means that for each $v \in T P$ , the following equation holds:

d π \circ (R_{g})_{*} (v) = d π (v)

In a specific point $p$

d π_{p} \circ (R_{g})_{*} (v) = d π_{p g^{- 1}} (v)

This commutative property encapsulates the equivariance of $d π$ under the right $G$ -action and indicates that the differential $d π$ is essentially "constant" along the $G$ -orbits in $P$ .

Alternative approach

Given a G-bundle $E \to M$ with standard fibre $S$ , the associated principal $G$ -bundle $P$ is, in a sense, a generalized frame bundle for the original bundle. To see this, observe that the principal $G$ -bundle can be obtained in the following way. Given a trivializing atlas ${(U_{α}, ϕ_{α})}$ of $E$ , we can define for every $m \in M$ the set

P_{m} := {{\tilde{ϕ}}_{α} (m) \circ g : α such that m \in U_{α}, g \in G}

Their elements are the different ways of express the fibre $E_{m}$ (maybe an "uncomfortable" space) in terms of the standard fibre $S$ (a "comfortable space), "preserving" the structure provided by $G$ . We can call them the $G$ -bases for $E_{m}$ (when $E_{m}$ is a vector space, $S$ is $R^{n}$ and $G = G L (n)$ the set $P_{m}$ is made of the different basis of $E_{m}$ . See the note basis and change of basis).

Observe that for $m \in U_{α β}$ it could happen

{\tilde{ϕ}}_{α} (m) \circ g = {\tilde{ϕ}}_{β} (m) \circ g^{'} \in P_{m}

and then must be $g = g_{α β} (m) g^{'} .$

So if we consider the space

P = ⨆_{m \in M} P_{m}

we can provide maps

\begin{array}{cccc} ψ_{α} : & U_{α} \times G & ⟶ & P \\ (m, g) & ⟼ & {\tilde{ϕ}}_{α} (m) \circ g \end{array}

such that it can be proven that $P$ is a principal $G$ -bundle with atlas ${(U_{α}, Ψ_{α})}$ .

(It can be proven with the final topology and the fact of $G$ being a Lie group...)

Components of a section

Thus, given a G-bundle and its associated principal $G$ -bundle constructed in this way, we can define the components of $v \in E_{m}$ with respect to any $ϕ \in P_{m}$ :

ϕ^{- 1} (v) \in S .

Then, any local section of $P$ can be called a $G$ -frame, and therefore every local trivialization $(U, ψ)$ is equivalent to a distinguished $G$ -frame:

p (m) = ψ (m, e), for m \in U

and $e$ being the identity in $G$ .
In some context they are called moving frames or choice of a gauge. In case that the G-bundle $E$ is the tangent bundle of the manifold, the concept of $G$ -frame is the usual frame.

Trivializations vs sections

The orbit of $G$ at $p \in P$ is just the fiber $π^{- 1} (π (p))$ and, moreover, $π^{- 1} (π (p)) ≅ G$ . We can imagine $π^{- 1} (π (p))$ like $G$ but without a distinguished point $e$ . For example, a circle bundle ( $G = S^{1} = {e^{i \cdot t}}$ ) has a fiber which could be rotated but no point is matched with the identity. That is, every fiber is a $G$ -torsor or principal homogeneous space.

In this context we can speak of the translation map (or gauge transformation)

τ : π^{- 1} (π (p)) \times π^{- 1} (π (p)) \to G .

Given $q, q^{'} \in π^{- 1} (π (p)),$ since the action is free, there exist one and only one $τ (p, p^{'}) \in G$ such that

p^{'} = p \cdot τ (p, p^{'})

and we will write $p^{- 1} p^{'} = τ (p, p^{'})$ . Think of it like the subtraction of points in affine space.

Proposition
Local sections of a $G$ -principal bundle are in one to one correspondence with local trivializations. $◼$

Proof
The idea is that sections mark a distinguished point in every fiber. Then it only rest to use the translation function defined above. $◼$

It turns out that a $G$ -principal bundle is trivial if and only if has a global section.

Connections

A fundamental idea is that of a principal connection on a principal bundle.

Group reduction

See extension and reduction of a principal bundle