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tangent vectors on the sphere[edit]


and by Poincaré-Hopf, there can be no nonvanishing vectors if the Euler characteristic doesn't vanish (confer hairy ball theorem). Thus, only for n odd are there nonvanishing vector fields. In this case, let

then there are 2b+8*a−1 independent vector fields. Thus it follows that if n is 0, 1, 3 or 7, there are n independent vector fields so the corresponding tangent bundles are trivial. These also give constructions for the 4 normed division algebras.


The criterion I like is to say a 1-parameter group of symmetries is "physical" if its generator does not vanish by virtue of the equations of motion, and "gauge" if the generator does vanish when the equations of motion hold. If our symmetry is "gauge" we should mod out by it; if it's "physical" we should not.

-John Baez

Usually people don't bother to define "forgetful functor" very precisely - like pornography, you're just supposed to know a forgetful functor when you see it.