caa a22 4500 465825176 CHVBK 20180323112147.0 cr unu---uuuuu 170327e19901201xx s 000 0 eng 10.1007/BF02112276 doi (NATIONALLICENCE)springer-10.1007/BF02112276 Alpers B. Zugspitzstr. 11, D-8035, Gauting, West Germany aut Note on angle-preserving mappings [Elektronische Daten] [B. Alpers] 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. Birkhäuser Verlag, 1990 Primary 51F20 nationallicence aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 1-7 0001-9054 40:1<1 1990 40 10 https://doi.org/10.1007/BF02112276 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02112276 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Alpers B. Zugspitzstr. 11, D-8035, Gauting, West Germany aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 1-7 0001-9054 40:1<1 1990 40 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer