A family of quadratic forms associated to quadratic mappings of spheres (Q1061198)

Let \((U,q_ U)\) and \((V,q_ V)\) be real positive definite quadratic spaces of dimension n and m respectively, n,m\(\geq 2\). Furthermore let \(S_ U\subset U\), \(S_ V\subset V\) denote the corresponding spheres and S(U,V) the set of quadratic maps f:U\(\to V\) such that \(q_ V(f(x))=q_ U(x)^ 2\) for all \(x\in U\). To every \(f\in S(U,V)\) there is associated a family of quadratic forms \(f_ e\) \((e\in S_ U)\) by the following definition: \(f_ e(z)=<f(z),f(e)>_ V\) where \(<.,.>_ V\) is the bilinear form corresponding to \(q_ V.\) The general form of a real quadratic mapping of spheres can be determined by studying the diagonalization of each form in an associated family of quadratic forms (Theorem 1.1). A certain set H(U,V) of mappings of spheres - socalled Hopf maps - are studied whose form resembles that of classical Hopf fibrations. The following Theorem 4.1 is proved: Let \(f\in H({\mathbb{R}}^ n,{\mathbb{R}}^ m)\). The family of forms \(\{f_ e\}e\in S^{n-1}\) are all nondegenerate only if \(n=2,4,8\) or 16.