Vector fields on spheres

In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.

Specifically, the question is how many linearly independent vector fields can be constructed on a sphere in N-dimensional Euclidean space. A definitive answer was made in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that there were at least ρ(N)-1 such fields (see definition below). Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found.

In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers ρ(N) determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of N odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem), so the case N even is an extension of that. Adams showed that the maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the (N − 1)-sphere is exactly ρ(N) − 1.

The construction of the fields is related to the real Clifford algebras, which is a theory with a periodicity modulo 8 that also shows up here. By the Gram–Schmidt process, it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis at each point.

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