Suppose $\Sigma^2\subset M^3$ is an one-sided, non-orientable embedding. Let $\pi:\tilde{\Sigma}\rightarrow\Sigma$ be the orientable double cover. Is it necessary that the pull back of normal bundle $\pi^*N\Sigma$ is a trivial bundle on $\tilde{\Sigma}$?
I know an answer here: Pullback of normal bundle by a covering map.. However, it does not work. It just shows that manifolds that is not simply connected admit a non-trivial line bundle. But our question here is a special line bundle, the above counterexample does not show that it is non-trivial.