So it is relatively easy to show that there exists only one smooth structure on 
the real line $\mathbb{R}$. So here are 2 natural questions:


Q1: Up to equivalence, is there only one real analytic structure on $\mathbb{R}$? If so,
then do we have a simple proof of that? 


Q2: Where can I find the simplest proofs that there exists only one smooth structure
on $\mathbb{R}^2$ and $\mathbb{R}^3$?


So I've heard that on $\mathbb{R}^4$ there are infinitly (in fact uncountably) many non-equivalent smooth structures, so what about real analytic strucutres? Is there some kind of moduli space of smooth structures on $\mathbb{R}^4$. if so, in how many ways is it possible to deform
a smooth structure into a real analytic one?



Smooth and analytic structures on low dimensional euclidian spaces
Regarding Q1, put an analytic Riemann metric on your 1-manifold. Integrating a unit speed vector field gives an analytic diffeomorphism to $\mathbb R$. Another way to prove analytic structures are unique is to notice the same argument that one uses to prove that the group of $C^k$-diffeomorphisms of $\mathbb R$ has the homotopy type of $\mathbb Z\_2$ works for analytic diffeomorphisms -- simply take the straight-line homotopy between your original diffeomorphism and either the identity or the negative identity, appropriately. 


Regarding Q2, I don't know much in the way of really simple proofs. But when $n=2$ you've got the Uniformization Theorem from complex analysis. That's relatively simple. 


