On mappings preserving orthogonality of nonsingular vectors

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1991

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Let q be a regular quadratic form on a vector space (V, F) and let f be the bilinear form associated with q. Then, {Mathematical expression} is the set of non-singular vectors of V, and for x, y ∈ {Mathematical expression}, {right parenthesis, less than}(x, y) {colon equals}f(x, y)2/(q(x) · q(y)) is the q-measure of (x, y), where {right parenthesis, less than}(x,y)=0 means that x, y are orthogonal. For an arbitrary mapping {Mathematical expression} we consider the functional equations {Mathematical expression} and we state conditions on (V, F, q) such that σ is induced by a mapping of a well-known type. In case of dim V ∈N{set minus}{0, 1, 2} ∧ {divides}F{divides} > 3, each of the assumptions (I), (II), (III) implies that there exist a ρ-linear injection ξ :V →V and a fixed λ ∈F{set minus}{0} such that Fxσ =Fxξ ∀x ∈ {Mathematical expression} and f(xξ, yξ)=λ · (f(x, y))ρ ∀x, y ∈V. Moreover, (II) implies ρ =idF ∧q(xξ) = λ ·q(x) ∀x ∈ {Mathematical expression}, and (III) implies ρ=idF ∧ λ ∈ {1,-1} ∧xσ ∈ {xξ, -xξ} ∀x ∈ {Mathematical expression}. Other results obtained in this paper include the cases dim V = 2 resp. dim V ∉N resp. {divides}F{divides} = 3. © 1991 Birkhäuser Verlag.

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