Can the twisting of mobius band be represented by a U (1) bundle?

Question

With the usual embedding of a mobius band, the strip is twisted by an angle pi, smoothly, as it goes round.I think this can be represented intrinsically, independent of the embedding, by attaching a non trivial U(1) bundle. Maybe a sub group of the SO(2) bundle that represents the orientation of an orthonormal frame. Is it true? And if so how do I express that properly?

Answer 1

There is no nontrivial $U(1)$-bundle over the circle. A better way to think about the Möbius band is as a (subset of a) nontrivial real line bundle over $S^1$, whose associated orthonormal frame bundle is a nontrivial principal $O(1)$-bundle.

It is true that the line bundle can be embedded into the trivial $\mathbb C$-bundle over $S^1$, corresponding to an embedding of the principal $O(1)$-bundle into the trivial $U(1)$-bundle. This seems to correspond to the angular twisting you're thinking about.