On Quadratic Forms Between Spheres

Abstract

For a quadratic form f: Sm → Sn, we first estimate the dimensions of its preimages and determine the cases when the upper bound is achieved. Combining results of Wood and Yiu, we then derive a complete classification of f when m≥ 2n-2. By using orthogonal multiplication, we also introduce a new construction of quadratic forms between spheres, generalizing the classical Hopf construction.