From: utzoo!decvax!yale-com!leichter
Newsgroups: net.math
Title: The Borsok-Ulam Theorem again
Article-I.D.: yale-com.958
Posted: Wed Feb 23 00:07:05 1983
Received: Wed Feb 23 05:03:17 1983

A couple of weeks ago, I mentioned the Borsok-Ulam theorm in this newsgroup.
Informally, the theorem says that at any given time, there is a pair of anti-
podal points on the Earth's surface which SIMULTANEOUSLY have the same tempe-
rature and air pressure.

Several people asked me for references or a proof.  I had neither.  I've been
thinking about it and I think I know how a proof would go.  It's been a long
time since I've done this sort of stuff, so what I have is hardly a complete
proof; but I think it's a correct start.

Formally, what the theorem says is this:  Let f,g:S2 -> R be two continuous
maps.  If x < S2 ('<' == 'element of', to use Lew Mammel's notation), write
x* for the antipodal point.  Then there is some x < S2 so that f(x) = f(x*)
and g(x) = g(x*).

Consider the pair  as a vector in the tangent plane to S2 at x,
for any x < S2.  Do this by picking some point x0, using  in
the tangent plane at x0; and then, for x != x0, use the vector 
in this SAME tangent plane - but then translate it to x using the connection
in the tangent bundle.  This should give you a continuous vector field F on
S2.  Similarly, we can define a vector field F* of pairs ,
starting at the same point x0.  Now consider the vector field F-F*.  This
is a continuous vector field on S2, so by the "billiard ball theorem", it
vanishes somewhere, say at X.  Thus,  = <0,0>,
so X is the point we wanted.

The use of the ANTIPODAL point is essential in constructing F*.  The tangent
planes at antipodal points are parallel, so the difference vector really
looks as I've written it.  If you use non-antipodal points, you still find
a vanishing point, but the relation it gives use is between "rotated"
versions of the vectors, so you can't separate out the f and g terms and
make independent statements about them.

Anyone wishing to formalize this - or pick holes in it - please feel free.
							-- Jerry
						decvax!yale-comix!leichter