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((Short description|How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere))
((more citations needed|date=May 2012))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 [[linear independence|linearly independent]] smooth nowhere-zero vector fields can be constructed on a [[hypersphere|sphere]] in <math>n</math>-dimensional [[Euclidean space]]. A definitive answer was provided in 1962 by [[Frank Adams]]. It was already known,<ref>((cite journal | first=I. M. | last=James| author-link=Ioan James | title=Whitehead products and vector-fields on spheres | journal=[[Proceedings of the Cambridge Philosophical Society]] | volume=53 | year=1957 | issue=4| pages=817–820 | doi=10.1017/S0305004100032928| s2cid=119646042))</ref> by direct construction using [[Clifford algebra]]s, that there were at least <math>\rho(n)-1</math> such fields (see definition below). Adams applied [[homotopy theory]] and [[topological K-theory]]<ref name=R603>((cite journal | first=J. F. | last=Adams | author-link=Frank Adams | title=Vector Fields on Spheres | journal=[[Annals of Mathematics]] | volume=75 | year=1962 | issue=3 | pages=603–632 | zbl=0112.38102 | doi=10.2307/1970213| jstor=1970213 ))</ref> to prove that no more independent vector fields could be found. Hence <math>\rho(n)-1</math> is the exact number of pointwise linearly independent vector fields that exist on an (<math>n-1</math>)-dimensional sphere.

==Technical details==
In detail, the question applies to the 'round spheres' and to their [[tangent bundle]]s: in fact since all [[exotic sphere]]s have isomorphic tangent bundles, the ''Radon–Hurwitz numbers'' <math>\rho(n)</math> determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere.  The case of <math>n</math> odd is taken care of by the [[Poincaré–Hopf index theorem]] (see [[hairy ball theorem]]), so the case <math>n</math> 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 (<math>n-1</math>)-sphere is exactly <math>\rho(n)-1</math>.

The construction of the fields is related to the real [[Clifford algebra]]s, which is a theory with a periodicity ''modulo''&nbsp;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''' <math>\rho(n)</math> occur in earlier work of [[Johann Radon]] (1922) and [[Adolf Hurwitz]] (1923) on the [[Hurwitz problem]] on [[quadratic form]]s.<ref name=R127>((cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 | page=127 ))</ref>  For <math>n</math> written as the product of an odd number <math>A</math> and a [[power of two]] <math>2^B</math>, write

:<math>B=c+4d, 0\leq c<4</math>.

Then<ref name=R127/>

:<math>\rho(n)=2^c+8d</math>.

The first few values of <math>\rho(2n)</math> are (from ((OEIS|id=A053381))):
:2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ...
For odd <math>n</math>, the value of the function <math>\rho(n)</math> is one.

These numbers occur also in other, related areas. In [[Matrix (mathematics)|matrix theory]], the Radon–Hurwitz number counts the maximum size of a linear subspace of the real <math>n\times n</math> matrices, for which each non-zero matrix is a [[similarity transformation (geometry)|similarity transformation]], i.e. a product of an [[orthogonal matrix]] and a [[scalar matrix]].  In [[quadratic form]]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==
((reflist))

* ((cite book | last=Porteous | first=I.R. | author-link=Ian R. Porteous | title=Topological Geometry | url=https://archive.org/details/topologicalgeome0000port | url-access=registration | publisher=Van Nostrand Reinhold | year=1969 | isbn=0-442-06606-6 | zbl=0186.06304  | pages=[https://archive.org/details/topologicalgeome0000port/page/336 336–352] ))
* ((cite web | first=H.R. | last=Miller | title = Vector fields on spheres, etc. (course notes) | url=https://ocw.mit.edu/courses/mathematics/18-915-graduate-topology-seminar-kan-seminar-fall-2014/math-talks/MIT18_915F14_Steenrod.pdf | access-date = 10 November 2018 ))

((DEFAULTSORT:Vector Fields On Spheres))
[[Category:Differential topology]]
[[Category:Theorems in topology]]

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