Generalities on bundles

Some definitions and facts:

A fibered manifold consists of a pair of manifolds $M$ and $E$ together with a surjective submersion

π : E ⟶ M,

see @saunders1989geometry page 2.
By the canonical submersion theorem, we can use in $E$ adapted coordinate systems, that is, if $d i m (E) = q + p$ and $d i m (M) = q$ we can use in $E$ coordinates

(x^{i}, u^{a}) : U \subseteq E ⟶ R^{q + p}; i = 1 \dots q, a = 1 \dots p

such that for $a, b \in U$ with $π (a) = π (b)$ we have

x^{i} (a) = x^{i} (b)

A fibre (or fiber) bundle is a fibered manifold such that every point $p$ has a local trivialization, that is, a triple $(W_{p}, F_{p}, t_{p})$ being $W_{p}$ a neighbourhood of $p$ , $F_{p}$ a manifold and

t_{p} : π^{- 1} (W_{p}) ⟶ W_{p} \times F_{p}

a diffeomorphism such that

p r_{1} \circ t_{p} = {π |}_{π^{- 1} (W_{p})}

It can be shown that $F_{p}$ is always the same, let's say $F$ (see @saunders1989geometry page 7), and is called the typical fibre.

Alternative approach: the G-bundles.

Special type of bundle: natural bundles.