leichter (02/23/83)
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 <f(x),g(x)> as a vector in the tangent plane to S2 at x,
for any x < S2. Do this by picking some point x0, using <f(x0),g(x0)> in
the tangent plane at x0; and then, for x != x0, use the vector <f(x),g(x)>
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 <f(x*),g(x*)>,
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, <f(X) - f(X*),g(X) - g(X*)> = <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