On quadratic forms between spheres (Q1386686)

The author estimates the dimension of preimages for quadratic forms, where the preimage of a quadratic form \(f:S^m\rightarrow S^n\) for a point \(q\) in the image of \(f\) is the intersection of \(S^n\) with a linear subspace. For the multiplicity of \(f\) \(d=\max \{\dim f^{-1}(f(p))\mid p\in S^m\}\) it is shown that \(\max\{0, m-n\}\leq d\leq \min\{ m-1,n-1\}\) if \(f\) is nonconstant. The author then determines the case when the upper bound is reached. In this case up to isometries \(f\) is a Hopf construction when \(d=n-1\) and \(f=i\circ f_\lambda\) (where \(f_\lambda :S^n\rightarrow S^{n+1}\) is a \(\lambda\)-construction and \(i:S^{m+1}\rightarrow S^n\) denotes the totally geodesic inclusion map) when \(d=m-1<n-1\). By combining the results from \textit{R. Wood} [Invent. Math. 5, 163-168 (1968; Zbl 0204.23805)] and \textit{Paul Y. H. Yiu} [Math. Proc. Camb. Philos. Soc. 100, 493-504 (1986; Zbl 0613.55009)] and by applying the representation theory of Clifford algebras it is shown that if \(f:S^{2n-2}\rightarrow S^n\) is a nonconstant quadratic form then \(n\in \{ 2,4,8\}\) and \(f\) is the restriction of a Hopf fibration to a great \((2n-2)\)-sphere.