An analytic method for the embedding of the Euclidean and pseudo-Euclidean geometries

It is known that an $n$-dimensional geometry of maximum mobility admits a group of motions of dimension $n(n+1)/2$. Many of these geometries are well-known, for example, the Euclidean and pseudo-Euclidean geometries. Such geometries are phenomenologically symmetric; i.e., their metric properties are equivalent to their group properties. In this paper we consider the examples of the two-dimensional Euclidean and pseudo-Euclidean geometries to develop an analytical method for their embedding. More exactly, we search for all possible functions of the form $f=f((x_i-x_j)^2\pm(y_i-y_j)^2,z_i,z_j)$, where, for example, $x_i, y_i, z_i$ are the coordinates of a point $i$. It turns out that there exist only the following embeddings: $f=(x_i-x_j)^2\pm(y_i-y_j)^2+(z_i-z_j)^2$ and $f=[(x_i-x_j)^2\pm(y_i-y_j)^2]\exp(2z_i+2z_j)$. Note that we obtain not only the well-known three-dimensional geometries (Euclidean and pseudo-Euclidean) but also less known geometries (three-dimensional special extensions of the two-dimensional Euclidean and pseudo-Euclidean geometries). It is found that all these geometries admit six-dimensional groups of motions. To solve the formulated problem, according to the condition of local invariance of the metric function, we write the functional equation
$$2[(x_i-x_j)(X_1(i)-X_1(j))+\epsilon(y_i-y_j)(X_2(i)-X_2(j))]\frac{\partial f}{\partial\theta}+X_3(i)\frac{\partial f}{\partial z_i}+X_3(j)\frac{\partial f}{\partial z_j}=0,$$
where all the components are analytic functions. This equation is expanded in a Taylor series and the coefficients of the expansion at identical products of powers of the variables are compared. This task is greatly simplified by using the Maple 15 computing environment. The obtained results are used to write differential equations, which are then integrated to find solutions to the embedding problem formulated earlier.