I am going through Nakahara's book "Geomety, topology and physics" on my own to prepare for my bachelor thesis. I got stuck on vector bundles, specially the local trivialization.
If I have a local trivialization, $\phi: U_i\times F\rightarrow \pi^{-1}(U_i)$, where $\pi: E\rightarrow M $ is the projection between the total space and base space. What is then the difference between $\pi^{-1}(U_i)$ and $\phi(p,f)$?
Furthermore, the book dosen't really go into any detail how one should do the explicit calculation for the trivialization, I have tried to find information online but I could not find anything. I have tried to determine explicitly the local trivializations for an example in the book.
$$L:=\{(p,v)\in \mathbb{C}P^n\times\mathbb{C}^{n+1}|v=ap, a\in\mathbb{C} \} $$
I have been able to figure out that the fibre is a line in $\mathbb{C}^{n+1}$ over $ \mathbb{C}P^n$, the transition function is $t_{ij}=z_i/z_j$ and that the group structure is $U(1)$. But I have no idea how to explicitly determine the local trivialization.
Any help is appreciated.