From: greg@matching.math.ucdavis.edu (Greg Kuperberg)
Newsgroups: sci.math.research
Subject: Re: Real analytic manifolds
Date: 5 Jul 1998 16:05:31 -0700

In article <6nolp1$g1u$1@nnrp1.dejanews.com>,
 <solovay@math.berkeley.edu> wrote:
>I'd like a reference for the following theorem. Any real analytic manifold
>has  a real analytic embedding into some Euclidean space.
>
>As I recall the theorem also has a variant saying that real analytic vector
>bundles have "sufficiently many global sections". This can be interpreted as
>saying they arise from a real analytic map into some Grassmanian.

It's called the Morrey-Grauert theorem, and it is sometimes phrased as
saying that a smooth manifold has a unique real analytic structure.
This follows from an earlier and much easier theorem of Whitney that
two diffoemorphic real analytic manifolds which are both analytically
embeddable are isomorphic.

Here are the original references:

H. Grauert, On Levi's problem and the imbedding of real-analytic
manifolds, Ann. of Math. 68 (1958), 460-472.

C. B. Morrey, The analytic embedding of abstract real-analytic manifolds,
Ann. of Math. 68 (1958), 159-201.
-- 
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==============================================================================

From: Joerg Winkelmann <winkelma@pauli.math-lab.unibas.ch>
Newsgroups: sci.math.research
Subject: Re: Real analytic manifolds
Date: Mon, 06 Jul 1998 15:26:01 +0100

solovay@math.berkeley.edu wrote:
> I'd like a reference for the following theorem. Any real analytic manifold
> has  a real analytic embedding into some Euclidean space.

see my previous posting
[Reaffirmed the Grauert citation listed above -- djr]

> As I recall the theorem also has a variant saying that real analytic vector
> bundles have "sufficiently many global sections". This can be interpreted as
> saying they arise from a real analytic map into some Grassmanian.

This is a direct consequence of the embedding theorem for manifolds:

Let E be a real-analytic vector bundle over a real-analytic manifold M.
Then the total space E is again a real analytic manifold ,
hence embeddable into some euclidean space.
Thus we may assume M \subset E \subset R^n.
Now E is, as a vector bundle, isomorphic to the normal bundle
of its zero-section, where the zero-section is considered as a
submanifold of the total space of the vector bundle.
It follows that M can be embedded into R^n such that E is
isomorphic to the normal bundle of some larger submanifold of R^n
which contains M. Using orthogonal complements it follows that
E is (in a real-analytic way) a direct summand of a trivial bundle
(=tangent bundle of the euclidean space)
over M. It follows that E is spanned by real-analytic global
sections.
Now, if a vector bundle E over a manifold M is spanned by  a finite-
dimensional vector space V of global sections, 
this yields a map from M to a Grassmann manifold which associates
to every point x in M the subvectorspace of those sections 
vanishing in x.
Furthermore E is the pull-back of the tautological bundle over
this Grassmann manifold.

Joerg Winkelmann
-- 
jwinkel@member.ams.org
Mathematisches Institut der Universitaet Basel 
http://www.cplx.ruhr-uni-bochum.de/~jw/