mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the discussion of vector fields on spheres was a classical problem of

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

, 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 provided in 1962 by Frank Adams. It was already known, by direct construction using

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...

s, that there were at least ρ(''N'')-1 such fields (see definition below). Adams applied

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...

and topological K-theory to prove that no more independent vector fields could be found. Hence ρ(''N'')-1 is the exact number of pointwise linearly independent vector fields that exist on an (N-1)-dimensional sphere.

Technical details

In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all

exotic sphere In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of al ...

s 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

s, 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

The Radon–Hurwitz numbers ''ρ''(''n'') occur in earlier work of Johann Radon (1922) and

Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...

(1923) on the Hurwitz problem on

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

s. 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 :''ρ''(''N'') = 2^''c'' + 8''d''. The first few values of ''ρ''(2''n'') are (from ): :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

s, 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

* * {{DEFAULTSORT:Vector Fields On Spheres Differential topology Theorems in topology