Hairy ball theorem

Remarks

Suppose $v$ has no zero, and put $w = v / \norm{v}$. Then $H(x, t) = \cos(\pi t) \, x + \sin(\pi t) \, w(x)$ lies on the sphere and is a homotopy from the identity to the antipodal map. The identity has degree $1$ and the antipodal map of $S^{2n}$ has degree $(-1)^{2n+1} = -1$, which contradicts the homotopy invariance of degree.

Example: model the wind on the Earth as a continuous tangent vector field on the sphere $S^2$. Then at any moment there is a point where the horizontal wind speed is zero.

The dimension must be even: on $S^{2n-1} \subset \R^{2n}$, the field $v(x_1, x_2, \ldots, x_{2n-1}, x_{2n}) = (-x_2, x_1, \ldots, -x_{2n}, x_{2n-1})$ is tangent and has no zero.

Going deeper

For a student

The theorem follows from the degree of the antipodal map. It is also the special case of the Poincaré–Hopf theorem for $S^{2n}$, since the Euler characteristic of $S^{2n}$ is $2 \neq 0$. It is the first case of the question how many linearly independent vector fields exist on a sphere, which Adams answered in the theorem on vector fields on spheres. It also implies that $S^{2n}$ carries no structure of a Lie group, since left translates of a nonzero tangent vector at the identity would give a nowhere vanishing vector field.