Quadratic Spherical Maps

Abstract

Let K be an infinite field of characteristic #2. For vector spaces X, Y over K of finite dimension, we defined a quadratic mapf: X→ Y by the following conditions :

$$f(ax) = {a^2}f(x),a \in K,x \in X$$

((5.1))

$$(x,y) \mapsto \frac{1}{2}[f(x + y) - f(x) - f(y)],x,y \in XS$$

((5.2))

is bilinear [see (1.2) and (1.3)]. We denoted by Q(X, Y) the set of all such maps. In what follows, we assume that X, Y are both nonsingular quadratic spaces with quadratic forms q _x , q _Y , respectively, and there are x∈X, y∈Y such that q _x (x) = qy(y) = 1. Therefore the unit spheres S _x , S _Y are nonempty:

$${S_X} = \{ x \in X;{q_X}(X) = 1\} ,{S_Y} = \{ y \in Y;{q_Y}(y) = 1\} $$

.