Vector fields on spheres
Encyclopedia
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 algebra
s.
Specifically, the question is how many linearly independent
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 algebra
s, 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.
s: in fact since all exotic sphere
s have isomorphic tangent bundles, the Radon–Hurwitz numbers compute 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. 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
2B. Write
Then
The construction of the fields is related to the real Clifford algebra
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.
(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 :
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the discussion of vector fields on spheres was a classical problem of differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, beginning with the hairy ball theorem
Hairy ball theorem
The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on an even-dimensional n-sphere. An ordinary sphere is a 2-sphere, so that this theorem will hold for an ordinary sphere...
, and early work on the classification of division algebra
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
s.
Specifically, the question is how many linearly independent
Linear independence
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
vector fields can be constructed on a sphere
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
in N-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. A definitive answer was made in 1962 by Frank Adams
Frank Adams
John Frank Adams FRS was a British mathematician, one of the founders of homotopy theory.-Life:He was born in Woolwich, a suburb in south-east London. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his Ph.D. from the University of...
. It was already known, by direct construction using Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
s, 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.
Technical details
In detail, the question applies to the 'round spheres' and to their tangent bundleTangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
s: in fact since all exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
s have isomorphic tangent bundles, the Radon–Hurwitz numbers compute 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
Hairy ball theorem
The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on an even-dimensional n-sphere. An ordinary sphere is a 2-sphere, so that this theorem will hold for an ordinary sphere...
), 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
Power of two
In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....
2B. Write
- B = c + 4d, 0 ≤ c < 4.
Then
- ρ(N) = 2c + 8d − 1.
The construction of the fields is related to the real Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
s, which is a theory with a periodicity modulo 8 that also shows up here. By the Gram–Schmidt process
Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn...
, it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
at each point.
Radon–Hurwitz numbers
The numbers ρ(n) are the Radon–Hurwitz numbers, so-called from the earlier work of Johann RadonJohann Radon
Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on calculus of variations .- Life :...
(1922) and Adolf Hurwitz
Adolf Hurwitz
Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...
(1923) in this area. A recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
is easy to give.
The first few values of ρ(2n) are given by :
- 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
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
and a scalar matrix. The classical results were revisited in 1952 by Beno Eckmann
Beno Eckmann
Beno Eckmann was a Swiss mathematician who was a student of Heinz Hopf.Born in Bern, Eckmann received his master's degree from Eidgenössische Technische Hochschule Zürich in 1931. Later he studied there under Heinz Hopf, obtaining his Ph.D. in 1941...
. They are now applied in areas including coding theory
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
.