TheoremProved

Hairy ball theorem

Statement

There is no continuous tangent vector field on the sphere $S^2$ that is nowhere zero; every continuous assignment of a tangent vector to each point of $S^2$ must vanish at at least one point.

Why is it true?

You cannot comb the hair flat all over a hairy ball without leaving at least one tuft standing up or a bald spot (a 'cowlick'). The same obstruction means that at every instant there must be at least one point on Earth's surface where the horizontal wind speed is exactly zero.

Proof sketch

By the Poincaré–Hopf index theorem, the sum of the indices of the isolated zeros of a continuous vector field on a compact surface equals the surface's Euler characteristic. Since $\chi(S^2) = 2 \neq 0$ , any continuous tangent vector field on $S^2$ must have at least one zero, since a nowhere-zero field would sum to index $0$ .

Stated by

stated 1885Henri Poincaré

Topics that use this theorem

Euler characteristic

Related theorems

Euler's polyhedron formula Brouwer fixed-point theorem Classification of closed surfaces

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

John Milnor (1978). Analytic proofs of the 'hairy ball theorem' and the Brouwer fixed point theorem
Victor Guillemin, Alan Pollack (1974). Differential Topology