Some Algebraic Geometry Aspects of Gravitational Theories with Covariant and Contravariant Connections and Metrics (GTCCCM) and Possible Applications to Theories with Extra Dimensions

TL;DR

This paper introduces an algebraic geometry approach to find new solutions to Einstein's equations in theories with covariant and contravariant metrics, with applications to extra-dimensional models and string theory.

Contribution

It proposes a novel algebraic geometry method for solving Einstein's equations as algebraic equations, including new relations in theories with extra dimensions.

Findings

Derived solutions for the length function l(x) in different compactification scenarios.

Established new parameter relations in string-inspired theories with extra dimensions.

Applied algebraic geometry to standard gravity and GTCCCM frameworks.

Abstract

On the base of the distinction between covariant and contravariant metric tensor components, an approach from algebraic geometry will be proposed, aimed at finding new solutions of the Einstein's equations both in GTCCCM and in standard gravity theory, if these equations are treated as algebraic equations. As a partial case, some physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. "length function" l(x) has been introduced and has been found as a solution of quasilinear differential equations in partial derivatives for two different cases, corresponding to "compactification + rescaling" and "rescaling + compactification" of the type I low-energy string theory action. New (although complicated) relations between the parameters in the action have been found, valid also for the standard approach in theories with extra dimensions.