Analytical embedding for geometries of constant curvature

V. A. Kyrov

Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 133-146.

Résumé

In various sections of modern mathematics and theoretical physics find their wide application of geometry of constant curvature. These geometries include spherical geometry, Lobachevsky geometry, de Sitter geometry. $n$-dimensional geometries of constant curvature are defined by metric functions that are invariants of motion groups of dimension $n(n + 1)/2$, therefore they are geometries of local maximum mobility. In this article, by the example of geometries of constant curvature, the embedding problem is solved, the essence of which is to find $ (n + 1) $ -dimensional geometries of local maximum mobility from $n$-dimensional geometries of constant curvature. We search for all functions of a pair of points of the form $f(A, B) = \chi(g(A,B),w_A,w_B) $ that define $(n + 1)$-dimensional geometries with motion groups of dimension $(n + 1)(n + 2)/2 $ by the well-known metric functions of $g(A, B)$ $n$-dimensional geometries of constant curvature. This problem reduces to solving functional equations of a special form in the class of analytic functions. The solution is sought in the form of Taylor series. To simplify the analysis of coefficients, the Maple 17 mathematical program package is used. The results of this embedding of $n$-dimensional geometries of constant curvature are $(n + 1)$-dimensional extensions of Euclidean and pseudo-Euclidean $n$-dimensional spaces. In addition to the main theorem, auxiliary statements of independent significance are proved.

Détail

Keywords: metric function, functional equation, geometry of constant curvature, group of motions.

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