"This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: there is no nonvanishing continuous tangent vector field on the sphere. Less briefly, if f is a continuous function that assigns a vector in R3 to every point p on a sphere, and for all p the vector f(p) is a tangent direction to the sphere at p, then there is at least one p such that f(p) = 0.
In fact from a more advanced point of view it can be shown that the sum at the zeroes of such a vector field of a certain 'index' must be 2, the Euler characteristic of the 2-sphere; and that therefore there must be at least some zero. In the case of the 2-torus, the Euler characteristic is 0; and it is possible to 'comb a hairy torus flat'."
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"This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: there is no nonvanishing continuous tangent vector field on the sphere. Less briefly, if f is a continuous function that assigns a vector in R3 to every point p on a sphere, and for all p the vector f(p) is a tangent direction to the sphere at p, then there is at least one p such that f(p) = 0.
In fact from a more advanced point of view it can be shown that the sum at the zeroes of such a vector field of a certain 'index' must be 2, the Euler characteristic of the 2-sphere; and that therefore there must be at least some zero. In the case of the 2-torus, the Euler characteristic is 0; and it is possible to 'comb a hairy torus flat'."
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