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.

中文翻译：

---

标量积的几何解析嵌入

摘要

我们解决了找到由非退化解析函数定义的所有\（（（n + 2）\）-维几何的问题

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

是尺寸\（（（n + 1）（n + 2）/ 2 \）的运动组的不变式。结果，我们有两个解决方案：预期标量积\（\ varepsilon _1x ^ 1_Ax ^ 1_B + \ cdots + \ varepsilon _ {n + 1} x ^ {n + 1} _Ax ^ {n + 1} _B + \ varepsilon w_Aw_B \）和意外的标量积 \（\ varepsilon _1x ^ 1_Ax ^ 1_B + \ cdots + \ varepsilon _ {n + 1} x ^ {n + 1} _Ax ^ {n + 1} _B + w_A + w_B \）。问题的解被简化为一种特殊的函数方程的解析解。