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Register a new userElliptic Curves and Algebraic Geometry Approach in Gravity Theory III. Uniformization Functions for a Multivariable Cubic Algebraic Equation
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Bogdan Georgiev Dimitrov
Publication date
2009
fields
Physics
and research's language is
English
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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).
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Bogdan G. Dimitrov (Bogoliubov Laboratory of Theoretical Physics
, n Joint Institute for Nuclear Research
, Dubna
2009
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.
168 -
Bogdan G. Dimitrov (Bogoliubov Laboratory of Theoretical Physics
, n Joint Institute for Nuclear Research
, Dubna
2009
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.
The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng [Z. B. Wu and J. Y. Zeng, Phys. Rev. A 62,032509 (2000)]. We find the similar properties in the responding systems in a spherical space, whose dynamical symmetries are described by Higgs Algebra. There exists a conserved aphelion and perihelion vector, which, together with angular momentum, constitute the generators of the geometrical symmetry group at the aphelia and perihelia points $(dot{r}=0)$.
In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {em ghosts} in the properties of the quantum mechanical solution. Bender and Mannheim cite{Bender 08 a} were successful in persuading the corresponding quantum operator to `give up the ghost. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physical community and in the process added further credit to the cause of advancement of the use of ${cal PT} $ symmetry. We present a case for the acceptance of the Pais-Uhlenbeck model in the context of Diracs theory by providing an Hamiltonian which is not quantum mechanically haunted. The essential point is the manner in which a fourth-order equation is rendered into a system of second-order equations. We show by means of the method of reduction of order cite {Nucci} that it is possible to construct an Hamiltonian which gives rise to a satisfactory quantal description without having to abandon Dirac.
A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure, the pairwise non-integrability condition and the bracket-generating condition, we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X subset {bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X subset {bf P}^N, dim X geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X to X$ is an isomorphism.
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