On Quadratic Forms Between Spheres

For a quadratic form f: S^m 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.