On permutations of central quadrics which preserve an inner distance (Q1895174)

Let \(V\) be a vector space over a commutative field \(F\) and assume \(\dim V \geq 4\), \(|F |\geq 4\). Let \(q : V \to F\) be a regular quadratic form with the associated bilinear form \(V \times V \to F\): \((x,y) \to xy : = q(x + y) - q(x) - q(y)\). The author considers a permutation \(\varphi\) of the central affine quadric \(Q : = \{x \in V \mid q(x) = 1\}\) such that \(xy = \mu \Leftrightarrow x^\varphi y^\varphi = \mu\) \(\forall x,y \in Q\) holds true, where \(\mu\) is a fixed element of \(F\). He proves that this mapping \(\varphi\) is metric induced resp. nearly metric induced in the sense that there exists a semilinear bijection \((\sigma, \rho)\): \((V,F) \to (V,F)\) such that \(q \circ \sigma = \rho \circ q\) and such that \(x^\varphi = x^\sigma\) \(\forall x \in Q\) resp. \(x^\varphi \in \{x^\sigma, -x^\sigma\} \forall x \in Q\), provided \(Q\) contains lines and the pair \((\mu,F)\) has additional properties if there are no planes in \(Q\). The cases \(\mu \neq 0\) and \(\mu = 0\) require different techniques. Some examples of the results obtained: \(\varphi\) is metric induced if \(\mu = 2\) or \(\mu = - 2\) (Theorems 1.1, 2.2); \(\varphi\) is nearly metric induced if \(\mu = 0\) and \(\text{ind} Q \geq 3\) (Theorem 3.10) or \(\text{ind} Q = 2\), \(|F |\geq 7\) (Theorem 3.12).