On angles and the fundamental theorems of metric geometry (Q910712)

A projective metric space is given by a vector space V over a commutative field K and a quadratic form q: \(V\to K\). The distance between non- singular points Kx and Ky is \(\delta (Kx,Ky):=f(x,y)^ 2q(x)^{- 1}q(y)^{-1}\) where \(f(x,y):=q(x+y)-q(x)-q(y).\) Main theorem. Let \(\Pi\) and \(\Pi '\) be positie metric spaces of dimensions \(\geq 2\) given by (V,K,q) and \((V',K',q')\). Let \(\delta\) and \(\delta '\) denote the distance mappings. Let \(\alpha\in K\) and \(\beta \in K'\) such that \(\delta (X,Y)=\alpha\) for some points X, Y of \(\Pi\) and \(\alpha\neq 0,4\). Furthermore, if \(| K| =2\) then assume that rad(V,f)\(\subseteq \ker nel q.\) Let \(\phi\) be a collineation of \(\Pi\) onto \(\Pi '\) such that \(\delta (X,Y)=\alpha \Leftrightarrow \delta (\phi (X),\phi (Y))=\beta\) for every pair of non-singular points X, Y. Then there is a semilinear bijection (\(\pi\),\(\sigma)\): (V,K)\(\to (V',K')\) and some \(\lambda\in K\setminus \{0\}\) such that \(\phi (Kv)=K'(\pi (v))\) for each \(v\in V\setminus \{0\}\) and \(q'(\pi (v))=\lambda \sigma (q(v))\) for every \(v\in V.\) A corresponding result in terms of affine spaces is proved where the distance function is replaced by an angle measure.