TITLE: Essential uniqueness  of the real-analytic structure on $\mathbb R$
QUESTION [14 upvotes]: 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.

REPLY [7 votes]: For Misha's comment and question 3 see the paper of David Minda Regular Analytic arcs and curves Colloq.Math 38(1977) no 1 73-82 .Regarding the vanishing theorem for real analytic manifolds this was proved by Henri Cartan Bulletin de la S.M.F tome 85 (1957) 77-99. Cartan assumed his manifolds were real analytically embedded in euclidean space . One just needs to know that the Complexification is Stein .This is due to Grauert's solution for the Levi problem.
  In case of one dimensional real analytic manifolds the complexification is Stein by the Runge type theorem of Behnke and Stein . You can also use the Behnke-Stein theorem to show that real analytic one manifolds have complete real analytic metrics and follow Milnor's proof.
   All proofs I know go through Complexifications .Existence of Complexification for real analytic manifold is due to Whitney and Bruhat