# 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 independent smooth nowhere-zero 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^{[1]}, 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^{[2]} to prove that no more independent vector fields could be found.

## Technical details[edit]

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.

## Radon–Hurwitz numbers[edit]

The **Radon–Hurwitz numbers** *ρ*(*n*) occur in earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) on the Hurwitz problem on quadratic forms.^{[3]} For *N* written as the product of an odd number *A* and a power of two 2^{B}, write

*B*=*c*+ 4*d*, 0 ≤*c*< 4.

Then^{[3]}

*ρ*(*N*) = 2^{c}+ 8*d*.

The first few values of *ρ*(2*n*) are (from (sequence A053381 in the OEIS)):

- 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ...

For odd *n*, the value of the function *ρ*(*n*) is one.

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 transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms; the classical results were revisited in 1952 by Beno Eckmann. They are now applied in areas including coding theory and theoretical physics.

## References[edit]

**^**James, I. M. (1957). "Whitehead products and vector-fields on spheres".*Proceedings of the Cambridge Philosophical Society*.**53**: 817–820.**^**Adams, J. F. (1962). "Vector Fields on Spheres".*Annals of Mathematics*.**75**: 603–632. doi:10.2307/1970213. Zbl 0112.38102.- ^
^{a}^{b}Rajwade, A. R. (1993).*Squares*. London Mathematical Society Lecture Note Series.**171**. Cambridge University Press. p. 127. ISBN 0-521-42668-5. Zbl 0785.11022.

- Porteous, I.R. (1969).
*Topological Geometry*. Van Nostrand Reinhold. pp. 336–352. ISBN 0-442-06606-6. Zbl 0186.06304. - Miller, H.R. "Vector fields on spheres, etc. (course notes)" (PDF). Retrieved 10 November 2018.