Fundamental Geometric Objects Research Articles

Point Correspondences between <i>N</i> + 1 Hypersurfaces of Projective Spaces and (<i>N</i> + 1)-Webs

For a correspondence in question we establish a sequence of fundamental geometrical objects of the correspondence and find invariant normalizations of the first and second orders of all hupersurfaces under the correspondence. We single out main tensors of the correspondence and establish a connection between the geometry of point correspondences between n + 1 hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs.

Partitioning two-dimensional mixed phase spaces

Hamiltonian systems typically exhibit a mixture of chaos and regularity, complicating any scheme to partition phase space and extract a symbolic description of the dynamics. In particular, the dynamics in the vicinity of stable islands can exhibit complicated topology that is qualitatively distinct from that away from the islands. We develop an approach to partition the chaotic phase space of a general dynamical system represented by a two-dimensional map (homeomorphism). This approach can accommodate mixed phase space structure with an arbitrarily high degree of accuracy. The partitioning scheme is built around networks of nested heteroclinic tangles—fundamental geometric objects that organize phase–space transport. These tangles can be used to progressively approximate the dynamics in the vicinity of stable island chains. The net result is a symbolic approximation to the dynamics in the chaotic sea, and an associated phase–space partition, which includes the influence of stable islands and which is approximately Markov.

Nonholonomic algebroids, Finsler geometry, and Lagrange-Hamilton spaces

We elaborate an unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructions are modelled on nonholonomic manifolds provided with nonintegrable distributions and related chains of exact sequences of submanifolds defining N-connections. We investigate the main properties of the Lagrange, Hamilton, Finsler-Riemann and Einstein-Cartan algebroids and construct and analyze exact solutions describing such objects.

Finsler branes and quantum gravity phenomenology with Lorentz symmetry violations

A consistent theory of quantum gravity (QG) at Planck's scale almost surely contains manifestations of Lorentz local symmetry violations (LV) which may be detected at observable scales. This can be effectively described and classified by models with nonlinear dispersions and related Finsler metrics and fundamental geometric objects (nonlinear and linear connections) depending on velocity/momentum variables. We prove that the trapping brane mechanism provides an accurate description of gravitational and matter field phenomena with LV over a wide range of distance scales and recovering in a systematic way the general relativity and local Lorentz symmetries. In contrast to the models with extra spacetime dimensions, the Einstein–Finsler-type gravity theories are positive with nontrivial nonlinear connection structure, nonholonomic constraints and torsion induced by generic off-diagonal coefficients of metrics, and determined by the fundamental QG and/or LV effects.

New discretization of complex analysis: The Euclidean and hyperbolic planes

Discretization of complex analysis on the plane based on the standard square lattice was started in the 1940s. It was developed by many people and also extended to the surfaces subdivided by the squares. In our opinion, this standard discretization does not preserve well-known remarkable features of the completely integrable system. These features certainly characterize the standard Cauchy continuous complex analysis. They played a key role in the great success of complex analysis in mathematics and applications. Few years ago, jointly with I. Dynnikov, we developed a new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially profound results were obtained for the Euclidean plane with an equilateral triangle lattice. Our approach preserves a lot of features of completely integrable systems. In the present work we develop a DCA theory for the analogs of an equilateral triangle lattice in the hyperbolic plane. This case is much more difficult than the Euclidean one. Many problems (easily solved for the Euclidean plane) have not been solved here yet. Some specific very interesting “dynamical phenomena” appear in this case; for example, description of boundaries of the most fundamental geometric objects (like the round ball) leads to dynamical problems. Mike Boyle from the University of Maryland helped me to use here the methods of symbolic dynamics.

Fractional almost Kähler–Lagrange geometry

The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kähler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a “prime” Lagrange (in particular, Finsler) generating function. We emphasize the importance of such constructions for deformation quantization of fractional Lagrange geometries and applications in modern physics.

Differentiable mappings of affine spaces into manifolds of m-planes in a multidimensional Euclidean space

In this paper we consider a differentiable mapping of a p-dimensional affine space into the differentiable manifold \( \mathfrak{M} \)N of all centered m-planes in an n-dimensional Euclidean space. We pay special attention to describing geometric images defined by a fundamental geometric object of the mentioned mapping.

BRANES AND QUANTIZATION FOR AN A-MODEL COMPLEXIFICATION OF EINSTEIN GRAVITY IN ALMOST KÄHLER VARIABLES

The general relativity theory is redefined equivalently in almost Kähler variables: symplectic form, θ[g], and canonical symplectic connection, [Formula: see text] (distorted from the Levi–Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metric g on a manifold V enabled with a necessary type nonholonomic 2 + 2 distribution. Such nonholonomic symplectic variables allow us to formulate the problem of quantizing Einstein gravity in terms of the A-model complexification of almost complex structures on V, generalizing the Gukov–Witten method [1]. Quantizing [Formula: see text], we derive a Hilbert space as a space of strings with two A-branes which for the Einstein gravity theory are nonholonomic because of induced nonlinear connection structures. Finally, we speculate on relation of such a method of quantization to curve flows and solitonic hierarchies defined by Einstein metrics on (pseudo) Riemannian spacetimes.

Einstein Gravity in Almost Kähler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics

We argue that the Einstein gravity theory can be reformulated in almost Kahler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation (NGT) on manifolds enabled with nonholonomic distributions is analyzed. There are considered some conditions when the fundamental geometric and physical objects are determined/ modified by nonholonomic deformations in general relativity or by contributions from Ricci flow theory and/or quantum gravity. We prove that in such NGT, for certain classes of nonholonomic constraints, there are modelled effective Lagrangians which do not develop instabilities. It is also elaborated a linearization formalism for anholonomic NGT models and analyzed the stability of stationary ellipsoidal solutions defining some nonholonomic and/or nonsymmetric deformations of the Schwarzschild metric. We show how to construct nonholonomic distributions which remove instabilities in NGT. Finally we conclude that instabilities do not consist a general feature of theories of gravity with nonsymmetric metrics but a particular property of certain models and/or classes of unconstrained solutions.

Quenched, minisuperspace, bosonicp-brane propagator

We borrow the minisuperspace approximation from Quantum Cosmology and the quenching approximation from QCD in order to derive a new form of the bosonic p-brane propagator. In this new approximation we obtain an exact description of both the collective mode deformation of the brane and the center of mass dynamics in the target spacetime. The collective mode dynamics is a generalization of string dynamics in terms of area variables. The final result is that the evolution of a p-brane in the quenched-minisuperspace approximation is formally equivalent to the effective motion of a particle in a spacetime where points as well as hypersurfaces are considered on the same footing as fundamental geometrical objects. This geometric equivalence leads us to define a new tension-shell condition that is a direct extension of the Klein-Gordon condition for material particles to the case of a physical p-brane.

A new geometrical gravitational theory

A geometrical gravitational theory based on the connection Γ ={ } + δ β α ∂γ lnϕ + δ γ α ∂β lnϕ −g βγ∂α ln ψ is developed. The field equations for the new theory are uniquely determined apart from one unknown dimensionless parameter Ω2. The geometry on which our theory is based is an extension of the Weyl geometry, and by the extension the gravitational coupling constant and the gravitational mass are made to be dynamical and geometrical. The fundamental geometrical objects in the theory are a metricg μν and two gauge scalarsϕ andψ. Physically the gravitational potential corresponds tog μν in the same way as in general relativity, the gravitational coupling constant toψ −2, and the gravitational mass tou(ϕ, ψ), which is a coscalar of power −1 algebraically made ofϕ andψ. The theory satisfies the weak equivalence principle, but breaks the strong one generally. We shall find outu(ϕ, ψ)=ϕ on the assumption that the strong one keeps holding good at least for bosons of low spins. Thus we have the simple correspondence between the geometrical objects and the gravitational objects. Since the theory satisfies the weak one, the inertial mass is also dynamical and geometrical in the same way as is the gravitational mass. Moreover, the cosmological term in the theory is a coscalar of power −4 algebraically made ofψ andu(ϕ, ψ), so it is dynamical, too. Finally we give spherically symmetric exact solutions. The permissible range of the unknown parameter Ω2 is experimentally determined by applying the solutions to the solar system.

Tetrad fields and metric tensor in the gauge theory of gravitation

The most relevant geometrical aspects of the gauge theory of gravitation are considered. A global definition of the tetrad fields is given and emphasis is placed on their role in defining an isomorphism between the tangent bundle of space-time and an appropriate vector bundle B associated to the gauge bundle. It is finally shown how to construct the fundamental geometrical objects on space-time, starting from B.

Torsion and curvature in extended supergravity

The first Bianchi identities are used to show, that in extended supergravity the curvature can be expressed in terms of the torsion and its covariant derivatives. The second Bianchi identities are shown to hold on account of the first ones. Thus all expressions containing the curvature may be written in terms of the torsion, which is the fundamental geometric object in supergravity.