Line bundles of the circle

Up to isomorphism, I think there exist only two line bundles of the circle: the trivial bundle (diffeomorphic to a cylinder) and a bundle that looks like to a Möbius band. Although it seems obvious geometrically I did not find a good argument to justify it. Do you have an idea?

Solution 1:

I'll show how the powerful (but sophisticated) theory of sheaves permits one to classify real line bundles on a paracompact topological space.
This will delight algebraic geometers and maybe motivate others to learn that theory.

Let $\mathcal C^*$ (resp. $\mathcal C^*_+)$ denote the sheaf of continuous nowhere zero functions (resp. sheaf of continuous positive functions) on a paracompact space $X$.
The exact sequence $ 0\to \mathcal C^*_+\to \mathcal C^*\stackrel {\text {sign}}\to \mathbb Z/2\mathbb Z\to0$ gives rise to a long exact sequence in cohomology of which a fragment is $$ \cdots \to H^1(X,\mathcal C^*_+) \to H^1(X,\mathcal C^*) \to H^1(X,\mathbb Z/2\mathbb Z)\to H^2(X,\mathcal C^*_+) \to\cdots$$
Now we have an isomorphism of sheaves $\mathcal C^*_+ \stackrel {\log} {\cong}\mathcal C$ and thus $\mathcal C^*_+$ is acyclic because $\mathcal C$ is acyclic (since it is a fine sheaf by paracompactness of $X$).
In particular $H^1(X,\mathcal C^*_+) = H^2(X,\mathcal C^*_+)=0$ so that the above cohomological fragment reduces to $ 0 \to H^1(X,\mathcal C^*) \to H^1(X,\mathbb Z/2\mathbb Z)\to 0 $ and since $H^1(X,\mathcal C^*)$ classifies line bundles on $X$ we get the result that line bundles on $X$ are classified by $H^1(X,\mathbb Z/2\mathbb Z)$.

In the differential geometry category the analogous result holds with $\mathcal C$ replaced by $\mathcal C^\infty$.
This yields the astonishing result that on a manifold each continuous line bundle has one and only one differential structure (up to isomorphism).

Finally, for the circle $H^1(S^1,\mathbb Z/2\mathbb Z)=\mathbb Z/2\mathbb Z$ and this proves your result.

Remark
The main reason I am posting this proof is for my record: I have never seen it in a reference and I want to be able to retrieve it in the probable case that I forget it!

Solution 2:

Line bundles are classified by the first Čech cohomology with coefficients in $\text{GL}^1(\mathbf R)$. By normalizing we can actually use coefficients in $\text{O}^1(\mathbf R)=\pm 1$. By using the usual covering of the circle, one sees immediately that this cohomology group is cyclic of order $2$, generated by the class of the Möbius strip.