In the case of the torusthe Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat". The hairy ball theorem of algebraic topology sometimes called the hedgehog theorem in Europe  states that there is no nonvanishing continuous tangent vector field on even-dimensional n -spheres. Hairy ball trophy an idealisation, take wind to be a two-dimensional vector: 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.
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Hairy ball theorem
With a View to Dynamical Systems. Note that the center with zero wind can be arbitrarily large or small. Like the hairy ball trophy hairs on the tennis ball, the wind will spiral around this zero-wind point - under our assumptions it cannot flow into or out of the point. Differential topology Fixed points mathematics Theorems in algebraic topology. In other projects Wikimedia Commons.
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This is a corollary of the hairy ball theorem. This section does not cite any sources. Views Read Edit View history. In other projects Wikimedia Commons.