On permutations of central quadrics which preserve an inner distance

Abstract

Letq be a regular quadratic form on a vector space (V,\(\mathbb{F}\)) and assume dimV ≥ 4 and ¦\(\mathbb{F}\)¦ ≥ 4. We consider a permutation ϕ of the central affine quadric\(\mathcal{F}\):= {x eV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of\(\mathbb{F}\) and where “·” is the scalar product associated withq. We prove that ϕ is induced (in a certain sense) by a semi-linear bijection (σ,ϱ): (V,\(\mathbb{F}\)) → (V,\(\mathbb{F}\)) such thatq o ϱo q, provided\(\mathcal{F}\) contains lines and the pair (μ,\(\mathbb{F}\)) has additional properties if there ar no planes in\(\mathcal{F}\). The cases μ,≠ 0 and μ = 0 require different techniques.