Hairy ball theorem
The theorem was first proved by Henri Poincaré for the 2-sphere in 1885,[4] and extended to higher even dimensions in 1912 by Luitzen Egbertus Jan Brouwer.
In the case of the torus, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat".
In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.
However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (equivalently, for every given vector).
There is a closely related argument from algebraic topology, using the Lefschetz fixed-point theorem.
By integrating a vector field we get (at least a small part of) a one-parameter group of diffeomorphisms on the sphere; and all of the mappings in it are homotopic to the identity.
It does suggest the correct statement of the more general Poincaré-Hopf index theorem.
The connection with the Euler characteristic χ suggests the correct generalisation: the 2n-sphere has no non-vanishing vector field for n ≥ 1.
Indeed it is easy to see that an odd-dimensional sphere admits a non-vanishing tangent vector field through a simple process of considering coordinates of the ambient even-dimensional Euclidean space
Whilst this does not preclude the possibility that there may still exist a tangent vector field to the even-dimensional sphere which does not vanish, the hairy ball theorem demonstrates that in fact there is no way of constructing such a vector field.
For example, rotation of a rigid ball around its fixed axis gives rise to a continuous tangential vector field of velocities of the points located on its surface.
[7] The hairy ball theorem may be successfully applied for the analysis of the propagation of electromagnetic waves, in the case when the wave-front forms a surface, topologically equivalent to a sphere (the surface possessing the Euler characteristic χ = 2).
At least one point on the surface at which vectors of electric and magnetic fields equal zero will necessarily appear.
[8] On certain 2-spheres of parameter space for electromagnetic waves in plasmas (or other complex media), these type of "cowlicks" or "bald points" also appear, which indicates that there exists topological excitation, i.e., robust waves that are immune to scattering and reflections, in the systems.
Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound.
The theorem also has implications in computer modeling (including video game design), in which a common problem is to compute a non-zero 3-D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task.
