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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... The hairy ball theorem of algebraic topology states that, in laymans terms, one cannot comb the hair on a ball in a smooth manner. This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: there is no nonvanishing... In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...

Solution of the problem

A definitive answer to the major question was made in 1962 by Frank Adams. He showed in all dimensions N that the conjectural number ρ(N) of linearly-independent vector fields on the (N − 1)-sphere in N-dimensional Euclidean space was correct. It was already known, by direct construction, that there were such fields; Adams applied homotopy theory to prove that no more independent vector fields could be found. Frank Adams may also refer to Frank Dawson Adams a Canadian geologist. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

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 In mathematics, an exotic sphere is a differential manifold M, such that from a topological point of view M is a sphere, but not from the point of view of its differential structure. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, the Poincaré-Hopf Theorem (also known as the Poincaré-Hopf index formula, Poincaré-Hopf index theorem, or Hopf index theorem) states: Let M be a compact differentiable manifold. ... The hairy ball theorem of algebraic topology states that, in laymans terms, one cannot comb the hair on a ball in a smooth manner. This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: there is no nonvanishing... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ...

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. Clifford algebras are a type of associative algebra in mathematics. ... In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ... In mathematics, an orthonormal basis of an inner product space V(i. ...

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. Johann Radon (December 16, 1887–May 25, 1956) was a mathematician born in Litomerice in Bohemia (now Czech Republic). ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... Recurrent redirects here; for the meaning of recurrent in contemporary hit radio, see Recurrent rotation. ...

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. Matrix theory is a branch of mathematics which focuses on the study of matrices. ... Several equivalence relations in mathematics are called similarity. ... In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ... Coding theory deals with the properties of codes, and thus with their fitness for a specific application. ... Theoretical physics is physics that employs mathematical models and abstractions rather than experimental processes. ...

Reference

• J. F. Adams, Vector Fields on Spheres, Annals of Math 75 (1962) 603-632.