History of Science and Mathematics Stack Exchange is a question and answer site for people interested in the history and origins of science and mathematics. It only takes a minute to sign up.
Sign up to join this community
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
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.
