Note on angle-preserving mappings

Summary

Let (V, K, q) be aq-regular metric vector space over a commutative field with quadratic formq and letA(V, K, q) be the corresponding affine-metric space. A metric collineation ofA(V, K, q) is a product of a translation and a semilinear bijection (σ ₁,σ ₂) (whereσ ₂ ∈ AutK) such that, for aλ ∈ K\{0}, we haveλqσ ₁ =σ ₂ q. For linesA + KB, A + KC whereA, B, C ∈ V\{X ∈ V∣q(X) = 0} we define an angle-measure < _q (A +KB, A +KC) ≔f(B, C)² q(B)⁻¹ q(C)⁻¹ wheref is the bilinear form corresponding toq. For a point tripleA, B, C we define < _q ABC≔ < _q (K(A − B),K(C − B)) whenever the right-hand side is defined. Now assume |K| > 5. In order to get minimal conditions for metric collineations we prove: Ifα ≠ 0, 4 is an occurring angle-measure and ifϕ is a permutation of the point set such that exactly the point triples with measureα are mapped to point triples with measureβ ≠ 0, 4, thenϕ is already a metric collineation.