In Topology courses one learns

An even dimensional sphere does not possess any continuous field of unit vectors

What is the importance of this result? I can't think of any applications off the top of my head.

For more explanation see J Milnor Analytic Proofs of the "Hairy Ball Theorem" and the Brouwer Fixed Point Theorem

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    $\begingroup$ This seems like a pure mathematics question, and likely should be on Math Stack Exchange. $\endgroup$ – HDE 226868 Aug 4 '16 at 22:46
  • $\begingroup$ it's a difficult question and mainly I am getting historical answers $\endgroup$ – john mangual Aug 5 '16 at 16:21

One important fact is that it shows that vector fields on the sphere are not the same as vector-valued functions, i.e. the tangent bundle of the sphere is not trivial. The tangent bundle of the circle is trivial. This is why we need sections of bundles to describe vector fields on manifolds, a more complicated notion than the vector-valued functions we can use to describe vector fields on $\mathbb{R}^n$.


At every time, there is a place on earth where there is no wind. But you could be as well in the eye of the tornado.

  • $\begingroup$ This is completely invalid. The earth's surface has plenty of holes such as through natural arches... $\endgroup$ – user21820 Sep 11 '16 at 10:50

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