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I would like to ask you some clarifications on an explicit example of local trivializations and transition functions of fibre bundles: namely on the [tex][-1,1]\hookrightarrow E\rightarrow S^1[/tex] bundle (which I guess is the simplest possible example).

Following Nakahara (chapter 9, example 9.1) we pick [tex]U_1=(0,2\pi)\,\,U_2=(-\pi,\pi)[/tex] as an open covering for the base space [tex]S^1[/tex] and label [tex]A=(0,\pi)\,\,B=(\pi,2\pi)[/tex] the intersection [tex]U_1\cap U_2[/tex].

Now Nakahara takes as local trivialization on A

[tex]\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)[/tex]

for [tex]\theta\in A \,\,t\in [-1,1][/tex], then he says that on the B section we have two possible choices, namely

[tex]1)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)[/tex]

[tex]2)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,-t)[/tex].

Now, my questions:

1) Shouldn't we have both this possibilities for the A sector as well? Does Nakahara simply not state them because they would not add anything to the example (you end up with either a cylinder or the moebius strip anyway :) )

2) Why, disregarding the specific sector (A or B), are we limited to the two choices aboce, i.e.

[tex]\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\pm t)[/tex]

?

I mean, couldn't I pick the choice

[tex]3)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\frac{t}{a})[/tex]

with [tex]a\in\mathcal R/\{0\}[/tex]??

In this way we should have for the transition function [tex]t_{12}(\theta):t\rightarrow \frac{t}{a}[/tex], so that with the final choice of

[tex]1)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)[/tex]

on the A sector and

[tex]3)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\frac{t}{a})[/tex]

on the B sector the bundle would have the structure group [tex]G=\{e,1/a\}[/tex] which is something different from the cylinder or the Moebious strip!!

Where am I mistaking??

Thanks to all of you for your time!