On angles and the fundamental theorems of metric geometry
type: Item
instance of: scholarly article
title: On angles and the fundamental theorems of metric geometry
zbMATH Open document ID: 0696.51007
DOI: 10.1007/BF03322626
author: Burkhard Alpers
published in: Results in Mathematics
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51F20
Mathematics Subject Classification ID: 51N30
zbMATH DE Number: 4140691
zbMATH Keywords: projective metric space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4140691