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 smooth nowhere-zero vector fields can be constructed on a sphere in
A definitive answer was provided in 1962 by Frank Adams.
Adams applied homotopy theory and topological K-theory[2] to prove that no more independent vector fields could be found.
is the exact number of pointwise linearly independent vector fields that exist on an (
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
determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere.
Adams showed that the maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the (
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.
In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real
The classical results were revisited in 1952 by Beno Eckmann.
They are now applied in areas including coding theory and theoretical physics.