A family of quadratic forms associated to quadratic mappings of spheres

AbstractThe 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. In particular, the eigenvalues provide a means for detecting maps which are of the Hopf type. When the eigenvalues are nonzero for every form in the family, the forms associated to ƒ:Sn→Sm give rise to a quadratic form on the tangent bundle of the unit sphere Sn. If ƒ is of the Hopf type, nondegeneracy of each form occurs only when n=1,3,7,15