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
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.
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Dedicated to Professor Walter Benz on the occasion of his 60th birthday
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Alpers, B., Schröder, E.M. On mappings preserving orthogonality of non-singular vectors. J Geom 41, 3–15 (1991). https://doi.org/10.1007/BF01258504
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DOI: https://doi.org/10.1007/BF01258504