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! *