I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that F is smoothly isomorphic…")
I do not know how to use the transition function in order to show that these bundles are isomorphic. I tried to find smooth trivializations of F, but I could not.
Edit: This is how Lee defined the Möbius bundle in his book. (Lee, page 105, example 5.2)
Best Answer
$F$ and the Möbius bundle $M$ are both rank-1 vector bundles over $S^1$. Let {$\tau_{ab}$} and {$\tilde{\tau_{ab}}$} denote the transition functions determined by the local trivializations of $F$ and $M$, respectively. Then $F$ and $M$ are smoothly isomorphic over $S^1$ iff for each $U_\alpha$ in the open cover of $S^1$, there exists a smooth map $\sigma_\alpha: U_\alpha \rightarrow GL(1,\mathbb{R})$ such that $$\tilde{\tau_{ab}}(p)= \sigma_\alpha(p)^{-1}\tau_{ab}(p)\sigma_\beta(p) $$
where $p$ is in the intersection of $U_\alpha$ and $U_\beta$.