Generalities on bundles

Some definitions and facts:

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

$$ \pi: E \longrightarrow M $$

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

$$ (x^i,u^a): U \subseteq E \longrightarrow \mathbb{R}^{q+p}; \textrm{ } i=1\ldots q, a=1\ldots p $$

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

$$ x^i(a)=x^i(b) $$

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

$$ t_{p}: \pi^{-1}\left(W_{p}\right) \longrightarrow W_{p} \times F_{p} $$

a diffeomorphism such that

$$ p r_{1} \circ t_{p}=\left.\pi\right|_{\pi^{-1}\left(W_{p}\right)} $$

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.

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Alternative approach: the G-bundles

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Author of the notes: Antonio J. Pan-Collantes

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