The Analytic Embedding of Geometries with Scalar Product

Abstract

We solve the problem of finding all \((n+2)\)-dimensional geometries defined by a nondegenerate analytic function

$$ \varphi (\varepsilon _1x^1_Ax^1_B+ \cdots +\varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B,w_A,w_B),$$

which is an invariant of a motion group of dimension \((n+1)(n+2)/2\). As a result, we have two solutions: the expected scalar product \(\varepsilon _1x^1_Ax^1_B+ \cdots +\varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B+\varepsilon w_Aw_B \) and the unexpected scalar product \(\varepsilon _1x^1_Ax^1_B+ \cdots +\varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B+w_A+w_B \). The solution of the problem is reduced to the analytic solution of a functional equation of a special kind.