The embedding of multidimensional special extensions of pseudo-Euclidean geometries

For modern science, the study of geometries of local maximum mobility is of particular importance, including Euclidean and pseudo-Euclidean geometries, symplectic geometry, and geometries of constant curvature. There is no complete classification of such geometries at the exist. The author of this article developed a method, called the method of embedding, which makes it possible to carry out such a classification. The essence of this method consists in finding functions that define geometries of dimension $n+1$ using known functions that define geometries of dimension $n$. In this case, the desired function as an argument contains a known function of dimension geometry $n$ and two more variables. In addition, the requirement of local invariance of this function with respect to the transformation group with $(n+1)(n+2)/2 $ parameters is imposed. Then the condition of local invariance is written, from which the functional-differential equation is derived to the desired function. In this paper, the solutions of this equation are sought analytically, in the form of Taylor row. The problem formulated for pseudo-Euclidean geometry has three classes of solutions (geometries of local maximum mobility): pseudo-Euclidean geometry, special expansion of pseudo-Euclidean geometries, geometry on the pseudo sphere. In this paper we pose the embedding problem for special extensions of pseudo-Euclidean geometries. It is proved that the solutions of this problem are not the geometries of the local maximum mobility.