I am currently studying Haefliger's paper "Homotopy and Integrablity". During the last chapter, he applies his theory of $\Gamma$-structures to analytic codimension $1$ foliations.
Throughout the argument he asserts "two analytic structures on $S^1$ are analytically isomorphic". I can imagine this is the case, just as there is a single holomorphic/analyitic (complex) structure on $S^2$; however, in my head, this is a consequence of the uniformization theorem... so I haven't been able to work out an analogous proof to the $1$-dimensional real analytic case.
My question is: Is this an obvious fact?
If so, where do I start? If NOT, is there any reference where I can look it up?