Elliptic Curves, Algebraic Geometry Approach in Gravity Theory and Uniformization of Multivariable Cubic Algebraic Equations

摘要

Based on the distinction between the covariant and contravariant metrictensor components in the framework of the affine geometry approach and the s.c."gravitational theories with covariant and contravariant connection andmetrics", it is shown that a wide variety of third, fourth, fifth, seventh,tenth- degree algebraic equations exists in gravity theory. This is importantin view of finding new solutions of the Einstein's equations, if they aretreated as algebraic ones. Since the obtained cubic algebraic equations aremultivariable, the standard algebraic geometry approach for parametrization oftwo-dimensional cubic equations with the elliptic Weierstrass function cannotbe applied. Nevertheless, for a previously considered cubic equation forreparametrization invariance of the gravitational Lagrangian and on the base ofa newly introduced notion of "embedded sequence of cubic algebraic equations",it is demonstrated that in the multivariable case such a parametrization isalso possible, but with complicated irrational and non-elliptic functions.After finding the solutions of a system of first - order nonlinear differentialequations, these parametrization functions can be considered also asuniformization ones (depending only on the complex uniformization variable z)for the initial multivariable cubic equation.