Elliptic Curves and Algebraic Geometry Approach in Gravity Theory III. Uniformization Functions for a Multivariable Cubic Algebraic Equation

摘要

The third part of the present paper continues the investigation of thesolution of the multivariable cubic algebraic equation for reparametrizationinvariance of the gravitational Lagrangian. The main result in this paperconstitutes the fact that the earlier found parametrization functions of thecubic algebraic equation for reparametrization invariance of the gravitationalLagrangian can be considered also as uniformization functions. These functionsare obtained as solutions of first - order nonlinear differential equations, asa result of which they depend only on the complex (uniformization) variable z.Further, it has been demonstrated that this uniformization can be extended totwo complex variables, which is particularly important for investigatingvarious physical metrics, for example the ADS metric of constant negativecurvature (Lobachevsky spaces).