- #1
lennyleonard
- 21
- 0
Hi everyone!
I would like to ask you some clarifications on an explicit example of local trivializations and transition functions of fibre bundles: namely on the [-1,1]\hookrightarrow E\rightarrow S^1 bundle (which I guess is the simplest possible example).
Following Nakahara (chapter 9, example 9.1) we pick U_1=(0,2\pi)\,\,U_2=(-\pi,\pi) as an open covering for the base space S^1 and label A=(0,\pi)\,\,B=(\pi,2\pi) the intersection U_1\cap U_2.
Now Nakahara takes as local trivialization on A
\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)
for \theta\in A \,\,t\in [-1,1], then he says that on the B section we have two possible choices, namely
1)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)
2)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,-t).
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.
\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\pm t)
?
I mean, couldn't I pick the choice
3)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\frac{t}{a})
with a\in\mathcal R/\{0\}??
In this way we should have for the transition function t_{12}(\theta):t\rightarrow \frac{t}{a}, so that with the final choice of
1)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)
on the A sector and
3)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\frac{t}{a})
on the B sector the bundle would have the structure group G=\{e,1/a\} which is something different from the cylinder or the Moebious strip!
Where am I mistaking??
Thanks to all of you for your time!
I would like to ask you some clarifications on an explicit example of local trivializations and transition functions of fibre bundles: namely on the [-1,1]\hookrightarrow E\rightarrow S^1 bundle (which I guess is the simplest possible example).
Following Nakahara (chapter 9, example 9.1) we pick U_1=(0,2\pi)\,\,U_2=(-\pi,\pi) as an open covering for the base space S^1 and label A=(0,\pi)\,\,B=(\pi,2\pi) the intersection U_1\cap U_2.
Now Nakahara takes as local trivialization on A
\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)
for \theta\in A \,\,t\in [-1,1], then he says that on the B section we have two possible choices, namely
1)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)
2)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,-t).
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.
\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\pm t)
?
I mean, couldn't I pick the choice
3)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\frac{t}{a})
with a\in\mathcal R/\{0\}??
In this way we should have for the transition function t_{12}(\theta):t\rightarrow \frac{t}{a}, so that with the final choice of
1)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,t)
on the A sector and
3)\,\,\,\phi_1^{-1}(u)=(\theta ,t)\,\,\text{and}\,\,\phi_2^{-1}(u)=(\theta ,\frac{t}{a})
on the B sector the bundle would have the structure group G=\{e,1/a\} which is something different from the cylinder or the Moebious strip!
Where am I mistaking??
Thanks to all of you for your time!