It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by an elementary argument as follows: using partitions of unity to construct a nowhere vanishing 1-form, integrate to obtain a diffeomorphism to a connected open submanifold of $\mathbb R$, and compose with some elementary diffeomorphism from the submanifold to $\mathbb R$.
Presented this way, the argument breaks down in the real-analytic case $k=\omega$ because partitions of unity are no longer available. Even in that case, the assertion is still true, since Grauert-Remmert have shown that $C^1$-diffeomorphic real-analytic manifolds are real-analytically diffeomorphic (assuming paracompactness, there being uncountably many inequivalent real-analytic structures on the long ray). However, this is a very difficult general result.
In the case at hand, it is not hard to see that the partitions of unity are merely a device for proving a cohomological vanishing theorem $$H^1({\mathbb R},{\mathcal E})=0$$ where ${\mathcal E}$ is the sheaf of germs of appropriately smooth real-valued functions. Indeed, consider a covering of $\mathbb R$ by open intervals $I_n$, each intersecting only its immediate predecessor and immediate successor, chosen small enough that each is real-analytically diffeomorphic to a standard interval (hence also to $\mathbb R$). Note that any collection of functions defined on the intersections $I_n\cap I_{n+1}$ yields a 1-cocycle. Since the $I_n$ are standard intervals, there exist everywhere positive 1-forms $\eta_n$ defined on $I_n$. Thus, there exist smooth functions $f_n$ defined on $I_n\cap I_{n+1}$ such that $\eta_{n+1}=(\exp f_n)\eta_n$ on that intersection. The vanishing theorem implies that the 1-cocycle {$f_n: n\in{\mathbb Z}$} is a 1-coboundary, that is, for some collection of functions $g_n$ defined on $I_n$, we have that $f_n$ is the restriction of $g_{n}-g_{n+1}$ to $I_{n}\cap I_{n+1}$. By construction, the 1-forms $(\exp g_n)\eta_n$ on $I_n$ are the restrictions of a globally defined positive 1-form $\eta$.
Question 1: How is the vanishing theorem established in the real-analytic case?
I imagine this must be well-known, but I've not been able to find such a discussion in the literature. Perhaps I am just looking in the wrong places. In any event, several years ago I put together such an argument. The idea is to consider each consecutive pair of intervals $I_n, I_{n+1}$ each slightly thickened to a complex neighborhood given by a smoothly bounded Jordan domain $D_n$, the intersection of consecutive neighborhoods being another Jordan domain. If the neighborhoods are small enough, the given functions $f_n$ extend complex-analytically to the intersections, the real part yielding values on $\partial(D_n\cap D_{n+1})$, which in turn (suitably extended by 0) yield a function on (say) $\partial D_n$ which we then extend harmonically, hence real-analytically, via the Poisson Integral formula.
Question 2: Is such an argument been written down in the literature, or otherwise well-known?
Of course, once we resort to patching suitable complex neighborhoods of the chart images, there is a quick and dirty proof via the Uniformization Theorem: it suffices to glue suitable real-symmetric neighborhoods to obtain a simply connected Riemann surface with an anticonformal involution whose fixed locus is the given 1-manfold.
Question 3: Is this surely well-known argument written down in the literature?
To be honest, I started out knowing the argument via Uniformization, and decided to see whether this could be reduced to more elementary considerations. The proposed argument to prove the vanishing theorem succeeds partially, but I was struck by the fact that I am still doing complex analysis, or at least potential theory. Maybe it's unreasonable to expect to be able to produce real-analytic functions without sneaking a peek into the complex plane.
