$G$ -bundles

An alternative approach to fibre bundles is that one of the $G$ -bundles:

Definition
Let $M$ , $S$ be manifolds. We say $E \overset{π}{\to} M$ is a $G$ -bundle over $M$ if we can find an open covering ${U_{α}}$ for $M$ , together with a collection of homeomorphisms (local trivializations)

ϕ_{α} : U_{α} \times S ⟶ π^{- 1} (U_{α})

such that

For $m \in U_{α}$ , we have a map

{\tilde{ϕ}}_{α} (m) (s) := ϕ_{α} (m, s)

for $s \in S$ such that is a diffeomorphism from $S$ to $π^{- 1} ({m})$ .

If $U_{α β} := U_{α} \cap U_{β} \neq 0$ the transition functions

g_{α β} (m) := {\tilde{ϕ}}_{α}^{- 1} (m) \circ {\tilde{ϕ}}_{β} (m)

satisfy $g_{α β} (m) \in G \subseteq Diff (S)$ .
$◼$

We remark the following observations:

For $m \in M$ , $E_{m} := π^{- 1} ({m})$ is called the fibre of $E$ over $m$ . Since $S$ is diffeomorphic to every fibre $E_{m}$ , is usually called the standard fibre.
The collection ${(U_{α}, ϕ_{α})}$ is called a trivializing atlas for $E \to M$ .
We have considered $G$ as a subgroup of $Diff (S)$ , but often is more convenient to think of $G$ as an abstract group with a faithful left action $λ$ over $S$ . We often denote the action by

λ (g_{α β} (m), s) = g_{α β} (m) \cdot s

or even by

λ (g_{α β} (m), s) = g_{α β} (m) (s)

when it does not result in confusion.

For an open set $U \subseteq M$ , any map

σ : U ⟶ E

such that $π \circ σ = i d_{M}$ is called a local section of the fibre bundle $E \overset{π}{\to} M$ . A local section $σ$ defined on a trivializing open set $U_{α}$ can be identified with a map

{\tilde{σ}}_{α} : U_{α} ⟶ S

such that $σ (x) = ϕ_{α} (x, {\tilde{σ}}_{α} (x))$ .

The best example for a $G$ -bundle is a vector bundle, where $S$ is a vector space $V$ and we take $G = GL (V)$ .
Another important case: principal bundles.

G-bundles

$G$ -bundles