Vector fields on spheres

<h2 id="technical-details">Technical details</h2>
In detail, the question applies to the 'round spheres' and to their <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundles</a>: in fact since all <a href="/facts/Exotic_sphere/3YKNwX1I">exotic spheres</a> have isomorphic tangent bundles, the Radon–Hurwitz numbers 
 
 
 
 ρ
 (
 n
 )
 
 
 {\displaystyle \rho (n)}
 
 determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 odd is taken care of by the <a href="/facts/Poincar%25C3%25A9%25E2%2580%2593Hopf_index_theorem/IV1aG75X">Poincaré–Hopf index theorem</a> (see <a href="/facts/Hairy_ball_theorem/pSX6zDBz">hairy ball theorem</a>), so the case 
 
 
 
 n
 
 
 {\displaystyle 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
 
 
 {\displaystyle n-1}
 
)-sphere is exactly 
 
 
 
 ρ
 (
 n
 )
 −
 1
 
 
 {\displaystyle \rho (n)-1}
 
.
The construction of the fields is related to the real <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebras</a>, which is a theory with a periodicity modulo 8 that also shows up here. By the <a href="/facts/Gram%25E2%2580%2593Schmidt_process/vRNpld3t">Gram–Schmidt process</a>, it is the same to ask for (pointwise) linear independence or fields that give an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> at each point.

<h2 id="radonhurwitz-numbers">Radon–Hurwitz numbers</h2>
The Radon–Hurwitz numbers 
 
 
 
 ρ
 (
 n
 )
 
 
 {\displaystyle \rho (n)}
 
 occur in earlier work of <a href="/facts/Johann_Radon/sJxXdkDo">Johann Radon</a> (1922) and <a href="/facts/Adolf_Hurwitz/urNgHh58">Adolf Hurwitz</a> (1923) on the <a href="/facts/Hurwitz_problem/KTh6Sbd2">Hurwitz problem</a> on <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> For 
 
 
 
 n
 
 
 {\displaystyle n}
 
 written as the product of an odd number 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and a <a href="/facts/Power_of_two/MA6WEpTt">power of two</a> 
 
 
 
 
 2
 
 B
 
 
 
 
 {\displaystyle 2^{B}}
 
, write

B
 =
 c
 +
 4
 d
 ,
 0
 ≤
 c
 <
 4
 
 
 {\displaystyle B=c+4d,0\leq c<4}
 
.
Then<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

ρ
 (
 n
 )
 =
 
 2
 
 c
 
 
 +
 8
 d
 
 
 {\displaystyle \rho (n)=2^{c}+8d}
 
.
The first few values of 
 
 
 
 ρ
 (
 2
 n
 )
 
 
 {\displaystyle \rho (2n)}
 
 are (from (sequence A053381 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>)):

2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ...
For odd 
 
 
 
 n
 
 
 {\displaystyle n}
 
, the value of the function 
 
 
 
 ρ
 (
 n
 )
 
 
 {\displaystyle \rho (n)}
 
 is one.
These numbers occur also in other, related areas. In <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix theory</a>, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 matrices, for which each non-zero matrix is a <a href="/facts/Similarity_transformation_(geometry)/Sbd6hEAN">similarity transformation</a>, i.e. a product of an <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrix</a> and a <a href="/facts/Scalar_matrix/tjIAHrE6">scalar matrix</a>. In <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>, the <a href="/facts/Hurwitz_problem/KTh6Sbd2">Hurwitz problem</a> asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by <a href="/facts/Beno_Eckmann/XevLI2Hh">Beno Eckmann</a>. They are now applied in areas including <a href="/facts/Coding_theory/SBqStoGL">coding theory</a> and <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>.

<ul><li><a href="/facts/Ian_R._Porteous/JV9YyBnB">Porteous, I.R.</a> (1969). <a href="https://archive.org/details/topologicalgeome0000port">Topological Geometry</a>. Van Nostrand Reinhold. pp. <a href="https://archive.org/details/topologicalgeome0000port/page/336">336–352</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-442-06606-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0186.06304">0186.06304</a>.</li>
<li>Miller, H.R. <a href="https://ocw.mit.edu/courses/mathematics/18-915-graduate-topology-seminar-kan-seminar-fall-2014/math-talks/MIT18_915F14_Steenrod.pdf">"Vector fields on spheres, etc. (course notes)"</a> (PDF). Retrieved 10 November 2018.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">James, I. M. (1957). "Whitehead products and vector-fields on spheres". Mathematical Proceedings of the Cambridge Philosophical Society. 53 (4): 817–820. doi:10.1017/S0305004100032928. S2CID 119646042. <a href="/wiki/Ioan_James" target="_blank">/wiki/Ioan_James</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Adams, J. F. (1962). "Vector Fields on Spheres". Annals of Mathematics. 75 (3): 603–632. doi:10.2307/1970213. JSTOR 1970213. Zbl 0112.38102. <a href="/wiki/Frank_Adams" target="_blank">/wiki/Frank_Adams</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. p. 127. ISBN 0-521-42668-5. Zbl 0785.11022. <a href="0-521-42668-5" target="_blank">0-521-42668-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. p. 127. ISBN 0-521-42668-5. Zbl 0785.11022. <a href="0-521-42668-5" target="_blank">0-521-42668-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Vector fields on spheres open-in-new

Vector fields on spheres