Quadratic forms between spheres and the non-existence of sums of squares formulae

Abstract

Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the typewhere are bilinear forms with real coefficients in and . Such an [r, s, n] formula is equivalent to a normed bilinear map satisfying . We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.

Harmonic morphisms and circle actions on 3- and 4-manifolds

“…Then ^ is a harmonic polynomial morphism defined by homogeneous polynomials of degree 2. By Lemma 4.5 and the result of Yiu [38], up to isometry ^ is the Hopf map ^ :…”

“…By Lemma 4.5, $|.s'm-i : S 171 ' 1 -> S^1 is a quadratic harmonic polynomial morphism between spheres. These have been classified by Yiu [38]. The Hopf fibrations are the only examples, which occur when n = 2,3,5,9 and m = 2(n -1).…”

Systems of quadratic inequalities

“…Moreover, the condition ker(ωp) = {0} for every ω ∈ S c−1 says also p |S 2c−1 : S 2c−1 → S c is a submersion. It is a well-known result (see [13]) that the preimage of a point through a quadratic map between spheres is a sphere, and thus p |S 2c−1 is the projection of a sphere-bundle between spheres, hence it must be a Hopf fibration.…”

Non-existence of quadratic harmonic maps of $S^{4}$ into $S^{5}$ or $S^{6}$

“…A spherical harmonic on S m of order p is an eigenfunction of the spherical Laplacian with eigenvalue λ p = p(p + m − 1). Gauchman, Toth, Lam, Tang, Ueno and Yiu have done much work on quadratic harmonic maps between spheres; see [5,6,7,8,9,10,11,12] for more details. So a quadratic harmonic map f : S m → S n is also called a λ 2 -eigenmap, and generally one can investigate a λ p -eigenmap.…”