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 The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, if ...

, and early work on the classification of

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...

s. Specifically, the question is how many

linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

smooth nowhere-zero vector fields can be constructed on a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

in ''N''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. A definitive answer was provided in 1962 by

Frank Adams John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. Life He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research ...

. 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 In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...

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 bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

s: 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

), 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 In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner prod ...

, it is the same to ask for (pointwise) linear independence or fields that give an

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...

at each point.

Radon–Hurwitz numbers

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

Johann Radon Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna). Life RadonBrigitte Bukovics: ''Biography of Johan ...

(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 In mathematics, the Hurwitz problem (named after Adolf Hurwitz) is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables. Description Th ...

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 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative ...

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 In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begi ...

, 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 In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...

and a

scalar matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...

. In

s, the

asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by

Beno Eckmann Beno Eckmann (31 March 1917 – 25 November 2008) was a Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry. Life Born in Bern, Eckmann received his master's degree fr ...

. They are now applied in areas including

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are stud ...

and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

References

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