Abstract
We solve the problem of finding all \((n+2)\)-dimensional geometries defined by a nondegenerate analytic function
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.
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ACKNOWLEDGMENTS
The author expresses his sincere gratitude to Professor Gennadiĭ Grigor’evich Mikailichenko for discussing the results.
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Kyrov, V.A. The Analytic Embedding of Geometries with Scalar Product. Sib. Adv. Math. 31, 27–39 (2021). https://doi.org/10.1134/S105513442101003X
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DOI: https://doi.org/10.1134/S105513442101003X