From: baez@math.ucr.edu (john baez)
Subject: Real-analytic diffeomorphisms of the sphere
Date: Sat, 15 Jan 2000 15:28:05 -0800 (PST)
Newsgroups: sci.math.research
Summary: Finding real-analytic maps with prescribe behaviour at some points

Here's a question about real-analytic diffeomorphisms of 
the 2-sphere whose answer would be "obviously yes" if I
were asking about plain old smooth diffeomorphisms.  I'm 
hoping the answer is still "yes" in the real-analytic case.

Given distinct points p_1,...,p_n and distinct points
q_1,...q_n in S^2, is there a real-analytic diffeomorphism 

f: S^2 -> S^2

with f(p_i) = q_i and with arbitrarily chosen derivatives
at the points p_i?  (By "arbitrarily chosen" I mean that
df at p_i can be any orientation-preserving 1-1 and onto 
linear map that we want.)  

Proving this would seem to require a bunch of technology that 
I'm hoping someone else has already developed.  Any references
to such technology would also be greatly appreciated.
==============================================================================

From: alanw@yuban.berkeley.edu (Alan Weinstein)
Subject: Re: Real-analytic diffeomorphisms of the sphere
Date: 16 Jan 2000 21:44:16 GMT
Newsgroups: sci.math.research

Look at the orbits of the group of analytic diffeomorphisms on the
connected space of n-tuples of oriented frames over n-tuples of
distinct points.  A simple infinitesimal calculation (approximate
smooth vector fields by analytic ones) shows that these orbits are
open, so the action on this connected space is transitive. Same goes
for any connected manifold.

Alan Weinstein
==============================================================================

From: greg@math.ucdavis.edu (Greg Kuperberg)
Subject: Re: Real-analytic diffeomorphisms of the sphere
Date: 22 Jan 2000 19:19:53 -0800
Newsgroups: sci.math.research

In article <200001152328.PAA16518@charity.ucr.edu>,
john baez <baez@math.ucr.edu> wrote:
>Given distinct points p_1,...,p_n and distinct points
>q_1,...q_n in S^2, is there a real-analytic diffeomorphism 
>
>f: S^2 -> S^2
>
>with f(p_i) = q_i and with arbitrarily chosen derivatives
>at the points p_i?  (By "arbitrarily chosen" I mean that
>df at p_i can be any orientation-preserving 1-1 and onto 
>linear map that we want.)  

It's certainly true, and I can give a construction which is in principle
quite explicit.  First choose a smooth map h with the desired properties.
By a form of Weierstrass approximation, there exists a *polynomial* f:R^3
-> R^3 which, when restricted to S^2, is C^1 close to h and satisfies
f(p_i) = q_i and also has the chosen derivatives.  Now compose this
polynomial with radial projection to S^2.
-- 
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