On mappings preserving orthogonality of non-singular vectors

Abstract

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then,$\dot V: = \{ z \in V|q(z) \ne 0\}$ is the set of non-singular vectors ofV, and forx, y ∈$\dot V$, ∢(x, y) ≔f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ∢(x,y)=0 means thatx, y are orthogonal.

For an arbitrary mapping$\sigma :\dot V \to \dot V$ we consider the functional equations

$$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$

and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N∖{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F∖{0} such thatF x ^σ =F x ^ξ ∀x ∈$\dot V$ andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ∀x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ∀x ∈$\dot V$, and (III) implies ρ=id _F ∧ λ ∈ {1,−1} ∧x ^σ ∈ {x ^ξ, −x ^ξ} ∀x ∈$\dot V$. Other results obtained in this paper include the cases dimV = 2 resp. dimV ∉N resp. ∣F∣ = 3.