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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?

edited Jan 20 2011 at 19:52

asked Jan 20 2011 at 13:56

Hugo Chapdelaine
3,337●3●18

	mathoverflow.net/questions/8789/… – Steven Gubkin Jan 20 2011 at 14:06
	Thanks a lot Steven for the link. So do you know of a simple proof that there exists only one real analytic structure on $\mathbb{R}$ which is compatible with its smooth structure? – Hugo Chapdelaine Jan 20 2011 at 18:59
	Grauert-Remmert is probably irrelevant for this simple case. – Hugo Chapdelaine Jan 20 2011 at 18:59
	It was my (offhand) impression that every topological group with underlying space an $\mathbb{R}$-manifold had a unique structure as a real-analytic Lie group (and that this is part of the Gleason-Montgomery-Zippin theory). This is at least one attractive uniqueness result. – Pete L. Clark Jan 20 2011 at 22:35

Ryan Budney · Accepted Answer · 2011-01-20 19:45:16Z UTC

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.

Smooth and analytic structures on low dimensional euclidian spaces

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