Prove the following “theorem of meteorology”: at any given moment in time, there exists a pair of antipodal points on the surface of the earth at which both the temperature and the barometric pressure are equal.
Solution
Model the surface of the earth by the sphere \(S^2\). Assume temperature and barometric pressure vary continuously over the surface at the fixed time in question. Define \[ F:S^2\to\mathbb R^2, \qquad F(x)=(T(x),P(x)), \] where \(T\) is temperature and \(P\) is pressure. By the Borsuk–Ulam theorem, there exists \(x\in S^2\) such that \[ F(x)=F(-x). \] Equality of the two coordinates gives \[ T(x)=T(-x),\qquad P(x)=P(-x). \] Thus the antipodal points \(x\) and \(-x\) have both the same temperature and the same barometric pressure.