Algebraische Kennzeichnung angeordneter Bachmann-Räume (Q1059277)

Let (V,f) be a vector space endowed with a symmetric bilinear form f. Assume the index of (V,f) is zero. Let \({\mathcal A}={\mathcal A}(V,f,k)\) be the affine space over (V,f) with an additional orthogonality relation for lines which depends on \(k\in K\), K commutative field of characteristic \(\neq 2\). A subspace T of \({\mathcal A}\) is a subset of \({\mathcal A}\) with the following properties: T contains zero and at least one other point; no two points in T are polar to each other; if T contains two points, then it also contains all corresponding Thales points; if T contains three collinear points, then it also contains the fourth reflection point. The author proves, if K is ordered and f is definite, then every convex metric subspace of \({\mathcal A}\) is either a generalized Dehn or a generalized Klein model. The description obtained by the author is of special interest in connection with F. Bachmann's characterization of absolute geometries since all convex metric subspaces of \({\mathcal A}\) are (up to isomorphism) exactly the ordered Bachmann spaces.