From: greg@math.ucdavis.edu (Greg Kuperberg) Subject: Re: Real analytic vs. smooth Date: 24 Dec 1999 08:35:13 -0800 Newsgroups: sci.math.research Keywords: parallelizability in smooth and analytic categories In article <3863646D.DDF959C8@math.duke.edu>, Robert Bryant wrote: >I'm trying to track down a proof of or counterexample to the following >statement: > > If M is a real analytic manifold that is parallelizable as a smooth >manifold, then it is parallelizable as a real analytic manifold. > >Using results of H. Cartan, I can easily prove this if I assume that M is >closed (= compact without boundary), but I'd like to know whether >or not it is true without having to make this assumption. More >generally, I'd like to know whether or not a real analytic vector >bundle over M is real analytically trivial if it is smoothly trivial. > > I expect that this is something that is well known, one way or >the other, but I don't know where to look. The theory of analytic straightening of a smooth manifold has a "soft" part, due to Whitney, and a "hard" part, due to Morrey and Grauert. The soft part says that every smooth n-manifold admits a real analytic structure, one that admits a real analytic embedding in some R^N. (Here N>n in general.) Weierstrass approximation is available; one corollary is that two embeddable r. a. structures are r. a. equivalent. The hard part, due to Morrey and Grauert, says that every r. a. structure on a smooth manifold is embeddable. So let G be a Lie group and let E be a G-bundle over M. If E is smoothly trivial then there is a diffeomorphism E -> GxM. I think it has an analytic approximation f [*]. f might not commute with the projections to M, but f^-1(s), where s is the identity section of GxM, is then an analytic section of E. [*] I have a slight problem here. I know that the approximation f can be a diffeomorphism when G and M are compact (but M is not required to be closed), because given an embeddings of E and GxM in R^N, you can just take f to be a polynomial composed with normal projection to GxM. I am less certain that such an f exists when M is not compact. But I would think that it must be so, because otherwise Grauert's theorem, which extends Morrey's proof to the non-compact case, would be too good to be true. -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the Math Archive Front at http://front.math.ucdavis.edu/ \/ * 10054+1912 articles and counting! *