Geodesic
Combinatorial Construction of Tangent Vector Fields on Spheres
Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 590-605

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For every odd $n$, on the sphere $S^n$, $\rho(n)-1$ linear orthonormal tangent vector fields, where $\rho(n)$ is the Hurwitz–Radon number, are explicitly constructed. For each $8\times8$ sign matrix, compositions for infinite-dimensional positive definite quadratic forms are explicitly constructed. The infinite-dimensional real normed algebras thus arising are proved to have certain properties of associativity and divisibility type.
Keywords: linear orthonormal tangent vector field, odd-dimensional sphere, composition of quadratic forms, Clifford algebra, Hurwitz–Radon theorem, Cayley number. 
A. A. Ohnikyan. Combinatorial Construction of Tangent Vector Fields on Spheres. Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 590-605. http://geodesic.mathdoc.fr/item/MZM_2008_83_4_a9/
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[1] J. F. Adams, “Vector fields on spheres”, Ann. of Math. (2), 75:3 (1962), 603–632 | DOI | MR | Zbl

[2] B. Eckmann, “Gruppentheoretischer Beweis des Satzes von Hurwitz–Radon über die Komposition quadratischer Formen”, Comment. Math. Helv., 15:1 (1942/3), 358–366 | DOI | MR | Zbl

[3] D. Khyuzmoller, Rassloennye prostranstva, Mir, M., 1970 | MR | Zbl

[4] P. Zvengrowski, “Canonical vector fields on spheres”, Comment. Math. Helv., 43:1 (1968), 341–347 | DOI | MR | Zbl

[5] D. Shapiro, Compositions of Quadratic Forms, de Gruyter Exp. Math., 33, Walter de Gruyter Co., Berlin, 2000 | MR | Zbl

[6] A. Hurwitz, “Über die Komposition der quadratischen Formen”, Math. Ann., 88:1–2 (1922), 1–25 | DOI | MR | Zbl

[7] J. Radon, “Lineare Scharen ortogonaler Matrizen”, Abh. Math. Sem. Univ. Hamburg, 1 (1922), 1–14 | DOI

[8] I. Kaplansky, “Infinite-dimensional quadratic forms admitting composition”, Proc. Amer. Math. Soc., 4:6 (1953), 956–960 | DOI | MR | Zbl

[9] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR | Zbl

[10] J. F. Adams, P. D. Lax, R. S. Phillips, “On matrices whose real linear combinations are nonsingular”, Proc. Amer. Math. Soc., 16:2 (1965), 318–322 | DOI | MR | Zbl

[11] P. D. Laks, R. S. Fillips, Teoriya rasseyaniya, Mir, M., 1971 | MR | Zbl

[12] J. F. Adams, P. D. Lax, R. S. Phillips, “Correction to “On matrices whose real linear combinations are nonsingular””, Proc. Amer. Math. Soc., 17:4 (1966), 945–947 | DOI | MR | Zbl

[13] A. A. Albert, “Absolute valued algebras”, Ann. of Math. (2), 48:2 (1947), 495–501 | DOI | MR | Zbl