Local trivial bundle
We call $\pi : \: E \rightarrow M$ (where $E$, $M$, $F$ are manifolds) a local trivial bundle with fiber $F$ if:
- $\pi$ is a surjection
- $\forall_{x \in M} \: \: \exists_{\text{neighborhood }U} \: : \: f_U : \: \pi^{-1} (U) \rightarrow U \times F, \quad p e_1 : \: U \times F \rightarrow U, \quad \pi : \: \pi^{-1}(U) \rightarrow U$, where $f_U$ is a diffeomorphism, and $p e_1$ is the projection on the first variable (?)
Global trivial bundle
It's a local trivial bundle, with the exception that $E = M \times F$
Vector bundle
$\pi : \: E \rightarrow M$ is a vector bundle, if:
$F = \mathbb{R}^n$
$\forall_{x \in M} \: \exists_{\text{Atlas }A\text{ na }M} \: : \: \text{Maps }(U, \phi), (V, \psi) \text{ in }x\text{ belong to }A$
Those definitions are very abstract for me, and I can't find a visual/geometric interpretation of what all of that means.
Could anyone (maybe on an easy example) show me this?