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

The third part of the present paper continues the investigation of the solution of the multivariable cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian. The main result in this paper constitutes the fact that the earlier found parametrization functions of the cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian can be considered also as uniformization functions. These functions are obtained as solutions of first - order nonlinear differential equations, as a result of which they depend only on the complex (uniformization) variable z. Further, it has been demonstrated that this uniformization can be extended to two complex variables, which is particularly important for investigating various physical metrics, for example the ADS metric of constant negative curvature (Lobachevsky spaces).

Elliptic Curves and Algebraic Geometry Approach in Gravity Theory II. Parametrization of a Multivariable Cubic Algebraic Equation

In a previous paper, the general approach for treatment of algebraic equations of different order in gravity theory was exposed, based on the important distinction between covariant and contravariant metric tensor components. In the present second part of the paper it has been shown that a multivariable cubic algebraic equation can also be parametrized by means of complicated, irrational and non-elliptic functions, depending on the elliptic Weierstrass function and its derivative. As a model example, the proposed before cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been investigated. This is quite different from the standard algebraic geometry approach, where only the parametrization of two-dimensional cubic algebraic equations has been considered. Also, the possible applications in modern cosmological theories has been commented.

Elliptic Curves and Algebraic Geometry Approach in Gravity Theory I.The General Approach

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third, fourth, fifth, sixth, seventh - degree algebraic equations exists in gravity theory. This fact, together with the derivation of the algebraic equations for a generally defined contravariant tensor components in this paper, are important in view of finding new solutions of the Einsteins equations, if they are treated as algebraic ones. Some important properties of the introduced in hep-th/0107231 more general connection have been also proved - it possesses affine transformation properties and it is an equiaffine one. Basic and important knowledge about the affine geometry approach and about gravitational theories with covariant and contravariant connections and metrics is also given with the purpose of demonstrating when and how these theories can be related to the proposed algebraic approach and to the existing theory of gravity and relativistic hydrodynamics.

Algebraic Geometry Approach in Gravity Theory and New Relations between the Parameters in Type I Low-Energy String Theory Action in Theories with Extra Dimensions

On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non - elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two-dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. 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 found as a solution of quasilinear differential equations in partial derivatives for two different cases of compactification + rescaling and rescaling + compactification. New physically important relations (inequalities) between the parameters in the action are established, which cannot be derived in the case $l=1$ of the standard gravitational theory, but should be fulfilled also for that case.

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

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 Einsteins 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.

Integral Geometry on the Lobachevsky Plane and the Conformal Wess-Zumino-Witten Model of Strings on an ADS3 Background

The main purpose of the report is to provide some argumentation that three seemingly distinct approaches of 1. Giveon, Kutasov and Seiberg (hep-th/9806194); 2. Hemming, Keski-Vakkuri (hep-th/0110252); Maldacena, Ooguri (hep-th/0001053) and 3. I. Bars (hep-th/9503205) can be investigated by applying the mathematical methods of integral geometry on the Lobachevsky plane, developed previously by Gelfand, Graev and Vilenkin. All these methods can be used for finding the transformations, leaving the Kac-Moody and Virasoro algebras invariant. The near-distance limit of the Conformal Field Theory of the SL(2, R) WZW model of strings on an ADS3 background can also be interpreted in terms of the Lobachevsky Geometry : the non - euclidean distance is conserved and the Lobachevsky formulae for the angle of parallelism is recovered. Some preliminary technique from integral geometry for inverting the modified integral representation for the Kac- Moody algebra has been demonstrated.