From: greg@zarquon.uchicago.edu (Greg Kuperberg) Newsgroups: sci.math.research Subject: Re: Real-analytic structures Date: Tue, 2 May 1995 18:57:27 GMT In article <3o5ao9$fq4@lyra.csx.cam.ac.uk>, John Baez wrote: >How different is the category of smooth (C^infinity) compact manifolds >from the category of real-analytic compact manifolds? Funny you should ask. My mom and I had to learn about this in Seifert Conjecture work; Palais and Thurston told us all about it. >An algebraic geometer I know once showed me a proof that >every compact smooth manifold admits a real-analytic >structure. This was proved by Whitney, and it does not require anything fancy like hypercohomology. It is true in the non-compact (but paracompact) case as well, although the compact case is simpler. M admits a smooth embedding in some R^n. M is the locus of a non-singular set of smooth equations. Approximating the equations with polynomials by the Weierstrass approximation theorem, one gets a manifold parallel to M which is not only real analytic but Nash. A Nash manifold is one with non-singular real algebraic gluing maps. >On the other hand, I have no idea if every compact smooth >manifold admits a *unique* real-analytic structure --- unique, >that is, up to smooth diffeomorphism. It's true. One can also use Whitney's methods to show that two difeomorphic real analytic manifolds are real-analytic equivalent provided that they both admit embeddings in Euclidean space, equivalently provided that global, real-valued, real analytic functions distinguish points. This preliminary result also holds true for any other category of functions and manifolds, or Nash category, with an implicit and inverse function theorem and a few other features. For example, it's true for Nash manifolds. The hard part is then showing that every real analytic manifold does indeed embed in Euclidean space. This is the Morrey-Grauert theorem. Grauert's proof, the more sophisticated one, takes as its starting point a complex analytic manifold which is a tubular neighborhood of the real analytic manifold. It then applies some hard stuff in several complex variables to prove the existence of many global, complex analytic functions on the complex manifold. The embedding theorem does not hold for Nash manifolds. There is the embeddable Nash circle, and then there is, e.g., R/Z. R/Z is not embeddable as a Nash manifold, because there are no periodic real algebraic functions on R. You get gazillions more Nash structures on the circle by taking R_+/x^Z for different x>1. If anyone can add good references to this summary, that would be most helpful.