Vector fields on spheres

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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 indepenent 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) such fields (see definition below). Adams applied homotopy theory to prove that no more independent vector fields could be found.

[edit] Technical details

In detail, the question applies to the 'round spheres' (not exotic spheres); and to their tangent bundles. 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. The maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the (N − 1)-sphere is computable by this formula: write N as the product of an odd number A and a power of two 2^B. Write

B = c + 4d, 0 ≤ c < 4.

Then

ρ(N) = 2^c + 8d − 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.

[edit] Radon-Hurwitz numbers

The numbers ρ(n) are the Radon-Hurwitz numbers, so-called from the earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) in this area. A recurrence relation is easy to give.

The first few values of ρ(2n) are given by (sequence A053381 in OEIS):

1, 3, 1, 7, 1, 3, 1, 8, 1, 3, 1, 7, 1, 3, 1, 9, ...

For odd n, the function ρ(n) is zero.

These numbers occur also in other, related areas. In matrix theory, the Radon-Hurwitz number counts the maximum size of a linear subspace of the real n×n matrices, for which each non-zero matrix is a similarity, i.e. a product of an orthogonal matrix and a scalar matrix. The classical results were revisited in 1952 by Beno Eckmann. They are now applied in areas including coding theory and theoretical physics.