Nonholonomic Frame Research Articles

Dark energy and dark matter configurations for wormholes and solitionic hierarchies of nonmetric Ricci flows and F(R,T,Q,Tm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(R,T,Q,T_{m})$$\\end{document} gravity

We extend the anholonomic frame and connection deformation method, AFCDM, for constructing exact and parametric solutions in general relativity, GR, to geometric flow models and modified gravity theories, MGTs, with nontrivial torsion and nonmetricity fields. Following abstract geometric or variational methods, we can derive corresponding systems of nonmetric gravitational and matter field equations which consist of very sophisticate systems of coupled nonlinear PDEs. Using nonholonomic frames with dyadic spacetime splitting and applying the AFCDM, we prove that such systems of PDEs can be decoupled and integrated in general forms for generic off-diagonal metric structures and generalized affine connections. We generate new classes of quasi-stationary solutions (which do not depend on time like coordinates) and study the physical properties of some physically important examples. Such exact or parametric solutions are determined by nonmetric solitonic distributions and/or ellipsoidal deformations of wormhole hole configurations. It is not possible to describe the thermodynamic properties of such solutions in the framework of the Bekenstein–Hawking paradigm because such metrics do not involve, in general, certain horizons, duality, or holographic configurations. Nevertheless, we can always elaborate on associated Grigori Perelman thermodynamic models elaborated for nonmetric geometric flows. In explicit form, applying the AFCDM, we construct and study the physical implications of new classes of traversable wormhole solutions describing solitonic deformation and dissipation of non-Riemannian geometric objects. Such models with nontrivial gravitational off-diagonal vacuum are important for elaborating models of dark energy and dark matter involving wormhole configurations and solitonic-type structure formation.

Nonholonomic Frames for Finsler space with Special (α,β)-metrics

In the present paper, we determine the nonholonomic Frames for Finsler space with special (α, β) -metrics of various type and also we observed the nonholonomic frames expesses as a Guage Transformation of Finsler metric.

Nonholonomic Frame for a Deformed $(\alpha,\beta )$-metric

Recently, in paper [14], we have introduced the following deformed $(\alpha, \beta)$-metric: $$ F_{\epsilon}(\alpha,\beta)=\frac{\beta^{2}+\alpha^{2}(a+1)}{\alpha}+\epsilon\beta $$ where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric; $\beta=b_{i}y^{i}$ is a 1-form, $\left|\epsilon\right|<2\sqrt{a+1}$ is a real parameter and $a\in \left(\frac{1}{4},+\infty\right)$ is a real positive scalar. The aim of this paper is to find the nonholonomic frame for this important kind of $(\alpha, \beta)$-metric and also to investigate the Frobenius norm for the Hessian generated by this kind of metric.

Nonassociative Geometry of Nonholonomic Phase Spaces with Star R‐flux String Deformations and (non) Symmetric Metrics

AbstractWe elaborate on nonassociative differential geometry of phase spaces endowed with nonholonomic (non‐integrable) distributions and frames, nonlinear and linear connections, symmetric and nonsymmetric metrics, and correspondingly adapted quasi‐Hopf algebra structures. The approach is based on the concept of nonassociative star product introduced for describing closed strings moving in a constant R‐flux background. Generalized Moyal‐Weyl deformations are considered when, for nonassociative and noncommutative terms of star deformations, there are used nonholonomic frames (bases) instead of local partial derivatives. In such modified nonassociative and nonholonomic spacetimes and associated complex/real phase spaces, the coefficients of geometric and physical objects depend both on base spacetime coordinates and conventional (co) fiber velocity/momentum variables like in (non) commutative Finsler‐Lagrange‐Hamilton geometry. For nonassociative and (non) commutative phase spaces modelled as total spaces of (co) tangent bundles on Lorentz manifolds enabled with star products and nonholonomic frames, we consider associated nonlinear connection, N‐connection, structures determining conventional horizontal and (co) vertical (for instance, 4+4) splitting of dimensions and N‐adapted decompositions of fundamental geometric objects. There are defined and computed in abstract geometric and N‐adapted coefficient forms the torsion, curvature and Ricci tensors. We extend certain methods of nonholonomic geometry in order to construct R‐flux deformations of vacuum Einstein equations for the case of N‐adapted linear connections and symmetric and nonsymmetric metric structures.

Decoupling and Integrability of Nonassociative Vacuum Phase Space Gravitational Equations With Star and R‐flux Parametric Deformations

AbstractWe prove that nonassociative star deformed vacuum Einstein equations can be decoupled and integrated in certain general forms on phase spaces involving real R‐flux terms induced as parametric corrections on base Lorentz manifold spacetimes. The geometric constructions are elaborated with parametric (on respective Planck, ℏ, and string, , constants) and nonholonomic dyadic decompositions of fundamental geometric and physical objects. This is our second partner work on elaborating nonassociative geometric and gravity theories with symmetric and nonsymmetric metrics, (non) linear connections, star deformations defined by generalized Moyal‐Weyl products, endowed with quasi‐Hopf algebra, or other type algebraic and geometric structures, and all adapted to nonholonomic distributions and frames. We construct exact and parametric solutions for nonassociative vacuum configurations (with nontrivial or effective cosmological constants) defined by star deformed generic off‐diagonal (non) symmetric metrics and (generalized) nonlinear and linear connections. The coefficients of geometric objects defining such solutions are determined by respective classes of generating and integration functions and constants and may depend on all phase space coordinates [spacetime ones, ; and momentum like variables, )]. Quasi‐stationary configurations are stated by solutions with spacetime Killing symmetry on a time like vector but with possible dependencies on momentum like coordinates on star deformed phase spaces. This geometric techniques of decoupling and integrating nonlinear systems of physically important partial differential equations (the anholonomic frame and connection deformation method, AFCDM) is applied in our partner works for constructing nonassociative and locally anisotropic generalizations of black hole and cosmological solutions and elaborating geometric flow evolution and classical and quantum information theories.

Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables

We study new classes of generic off-diagonal and diagonal cosmological solutions for effective Einstein equations in modified gravity theories (MGTs), with modified dispersion relations (MDRs), and encoding possible violations of (local) Lorentz invariance (LIVs). Such MGTs are constructed for actions and Lagrange densities with two non-Riemannian volume forms (similar to two measure theories (TMTs)) and associated bimetric and/or biconnection geometric structures. For conventional nonholonomic 2 + 2 splitting, we can always describe such models in Finsler-like variables, which is important for elaborating geometric methods of constructing exact and parametric solutions. Examples of such Finsler two-measure formulations of general relativity (GR) and MGTs are considered for Lorentz manifolds and their (co) tangent bundles and abbreviated as FTMT. Generic off-diagonal metrics solving gravitational field equations in FTMTs are determined by generating functions, effective sources and integration constants, and characterized by nonholonomic frame torsion effects. By restricting the class of integration functions, we can extract torsionless and/or diagonal configurations and model emergent cosmological theories with square scalar curvature, R2, when the global Weyl-scale symmetry is broken via nonlinear dynamical interactions with nonholonomic constraints. In the physical Einstein–Finsler frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions, which allows for a unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are “emergent universes” described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce massive gravity effects in two-measure theories. Finally, we study a reconstructing procedure for reproducing off-diagonal FTMT and massive gravity cosmological models as effective Einstein gravity or Einstein–Finsler theories.

A Distributed Two-Layer Framework for Teleoperated Platooning of Fixed-Wing UAVs via Decomposition and Backstepping

We propose a novel distributed control framework for teleoperated platooning of multiple three-dimensional (3D) fixed-wing unmanned aerial vehicles (UAVs), consisting of the following two layers: 1) virtual frame layer, which generates the target 3D nonholonomic motion of the virtual nonholonomic frames (VNFs) using the nonholonomic decomposition (D. J. Lee, 2010) and backstepping in such a way that the VNFs are to maintain the platoon formation in a distributed manner while respecting the directionality of the fixed-wing UAV; and 2) local control layer, which drives each fixed-wing UAV to track their respective VNF with their under-actuation and aerodynamic disturbance effects fully taken into account by using the backstepping and Lyapunov-based design techniques. Convergence and stability of salient aspects of each layer and their combination are theoretically established. Simulations with 25 fixed-wing UAVs and a haptic device are also performed to validate the theory with their multimedia provided at https://youtu.be/Z3Mo66KIsns.

Sigma Functions and Lie Algebras of Schrödinger Operators

In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $$g > 0$$ , a system of $$2g$$ multidimensional Schrodinger equations in magnetic fields with quadratic potentials was defined. Such systems are equivalent to systems of heat equations in a nonholonomic frame. It was proved that such a system determines the sigma function of the universal hyperelliptic curve of genus $$g$$ . A polynomial Lie algebra with $$2g$$ Schrodinger operators $$Q_0, Q_2, \dots, Q_{4g-2}$$ as generators was introduced. In this work, for each $$g > 0,$$ we obtain explicit expressions for $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ and recurrent formulas for $$Q_{2k}$$ with $$k>2$$ expressing these operators as elements of a polynomial Lie algebra in terms of the Lie brackets of the operators $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ . As an application, we obtain explicit expressions for the operators $$Q_0, Q_2, \dots, Q_{4g-2}$$ for $$g = 1,2,3,4$$ .

Lie Algebras of Heat Operators in a Nonholonomic Frame

We construct the Lie algebras of systems of $$2g$$ graded heat operators $$Q_0,Q_2,\dots,Q_{4g-2}$$ that determine the sigma functions $$\sigma(z,\lambda)$$ of hyperelliptic curves of genera $$g=1$$ , $$2$$ , and $$3$$ . As a corollary, we find that the system of three operators $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ is already sufficient for determining the sigma functions. The operator $$Q_0$$ is the Euler operator, and each of the operators $$Q_{2k}$$ , $$k>0$$ , determines a $$g$$ -dimensional Schrodinger equation with potential quadratic in $$z$$ for a nonholonomic frame of vector fields in the space $$\mathbb C^{2g}$$ with coordinates $$\lambda$$ . For any solution $$\varphi(z,\lambda)$$ of the system of heat equations, we introduce the graded ring $$\mathscr R_\varphi$$ generated by the logarithmic derivatives of $$\varphi(z,\lambda)$$ of order $$\ge 2$$ and present the Lie algebra of derivations of $$\mathscr R_\varphi$$ explicitly. We show how this Lie algebra is related to our system of nonlinear equations. For $$\varphi(z,\lambda)=\sigma(z,\lambda)$$ , this leads to a well-known result on how to construct the Lie algebra of differentiations of hyperelliptic functions of genus $$g=1,2,3$$ .

Lie Algebras of Heat Operators in a Nonholonomic Frame

Построены алгебры Ли систем из $2g$ градуированных операторов теплопроводности $Q_0,Q_2,…,Q_{4g-2}$, определяющих сигма-функции $\sigma(z,\lambda)$ гиперэллитических кривых рода $g=1,2$ и $3$. В качестве следствия получено, что системы из трех операторов $Q_0$, $Q_2$ и $Q_4$ уже достаточно, чтобы определить сигма-функции. Оператор $Q_0$ является оператором Эйлера, а каждый из операторов $Q_{2k}$, $k>0$, задает $g$-мерное уравнение Шрeдингера с квадратичным потенциалом по $z$ в неголономном репере векторных полей в $\mathbb C^{2g}$ с координатами $\lambda$. Для любого решения $\varphi(z,\lambda)$ системы уравнений теплопроводности мы вводим градуированное кольцо $\mathscr R_\varphi$, порожденное логарифмическими производными от функции $\varphi(z,\lambda)$ порядка не менее $2$ и в явном виде предъявляем алгебру Ли дифференцирований кольца $\mathscr R_\varphi$. Показана связь этой алгебры Ли с нашей системой нелинейных уравнений. В случае, когда $\varphi(z,\lambda)=\sigma(z,\lambda)$, это приводит к известному результату построения алгебры Ли дифференцирований гиперэллитических функций рода $g=1,2,3$. Библиография: 22 названия.

Аналитические и теоретико-числовые свойства двумерных сигма-функций

Обзор посвящен классическим и современным задачам, связанным с целой функцией $\sigma({\bf u};\lambda)$, которая определяется семейством неособых алгебраических кривых рода 2, где ${\bf u}= (u_1,u_3)$, $\lambda=(\lambda_4, \lambda_6, \lambda_8, \lambda_{10})$. Эта функция является аналогом сигма-функции Вейерштрасса $\sigma({{u}};g_2,g_3)$ семейства эллиптических кривых. Логарифмические производные порядка 2 и выше функции ${\sigma({\mathbf{u}};\lambda)}$ порождают поле гиперэллиптических функций от ${\mathbf{u}} = (u_1,u_3)$ на якобианах кривых с фиксированным значением вектора параметров $\lambda$. Мы рассматриваем три ряда Гурвица $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ и $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. Обзор посвящен теоретико-числовым свойствам функций $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ и $\mu_k(u_3;\lambda)$. Он включает самые последние результаты, доказательства которых использует тот фундаментальный факт, что функция $\sigma ({\mathbf{u}};\lambda)$ определяется системой четырех уравнений теплопроводности в неголономном репере шестимерного пространства.

A Uniqueness Theorem for the Two-Dimensional Sigma Function

We prove that the sigma functions of Weierstrass (g = 1) and Klein (g = 2) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of 2g linear differential heat equations in a nonholonomic frame (for a function of 3g variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For g =1, we give a complete description of the envelopes of holomorphy of such solutions.

KINEMATICS RENORMALIZATION OF ENERGY IN THE THEORY OF GRAVITY

In this paper, we study the change in the energy-momentum of a gravitational field when the general reference frame is changed. From a mathematical point of view, the general reference frame is described by the affine reference field on the space-time manifold, and the transitions between them form the gauge general linear group . Under the transformations of the group the Lagrangian of the theory, obtains a divergent term: its canonical transformation takes place. Consequently, the gravitational equations do not change, however, the total energy of the system and its superpotential are changes. This phenomenon is called the kinematic renormalization of energy and can be used to deprive the theory of singularities. The paper also examines individual cases of the selection of reference frames, in particular the so-called holonomic reference frames used in the general theory of relativity, as well as non-holonomic pseudoorthonormal reference frames. In the case of holonomic reference frames, it is possible to associate with them coordinate systems whose coordinate rays will set the reference frames. In contrast to the Euler coordinates , which number the points of space-time and whose change does not lead to a change in any physical quantities, the Lagrangian coordinates carry the physical content, since their change leads to a change in the physical reference frame, as a result of which certain physical quantities change, such as the energy-momentum of the system, and so on. In the case of coordinate transformations there are gauge translations in space-time in a fixed general reference frame belonging to the group . In the case of Lagrange coordinates transformations the transformations of holonomic reference frames occur, and such transformations belong to another group, namely the group , forming its subgroup, which is isomorphic to the group of diffeomorphisms due to restriction by holonomic reference frames. In the case of the identification of the Lagrangian coordinates with the Eulerian , the energy-momentum tensor becomes Einstein energy-momentum complex of gravitational field, and its superpotential becomes Freud superpotential. Obtained results open up wide opportunities for depriving the theory of gravity from singularities by choosing the appropriate general reference frames.

Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler\u2013Lagrange\u2013Hamilton geometry

We develop an axiomatic geometric approach and provide an unconventional review of modified/nonlinear gravity theories, MGTs, with modified dispersion relations, MDRs, encoding Lorentz invariance violations, LIVs, classical and quantum random effects, anisotropies etc. There are studied Lorentz–Finsler like theories elaborated as extensions of general relativity, GR, and quantum gravity, QG, models and constructed on (co)tangent Lorentz bundles, i.e. (curved) phase spaces or locally anisotropic spacetimes. An indicator of MDRs is considered as a functional on various type functions depending on phase space coordinates and physical constants. It determines respective generating functions and fundamental physical objects (generalized metrics, connections and nonholonomic frame structures) for relativistic models of Finsler, Lagrange and/or Hamilton spaces. We show that there are canonical almost symplectic differential forms and adapted (non)linear connections which allow us to formulate equivalent almost Kähler–Lagrange/–Hamilton geometries. This way, it is possible to unify geometrically various classes of (non)commutative MGTs with locally anisotropic gravitational, scalar, non-Abelian gauge field, and Higgs interactions. We elaborate on theories with Lagrangian densities containing massive graviton terms and bi-connection and bi-metric modifications which can be modelled as Finsler–Lagrange–Hamilton geometries. An example of short-range locally anisotropic gravity on (co)tangent Lorentz bundles is analysed. We conclude that a large class of such MGTs admits a self-consistent causal axiomatic formulation which is similar to GR but involving generalized (non)linear connections, Finsler metrics and adapted frames on phase spaces. Such extensions of the standard model of particle physics and gravity offer a comprehensive guide to classical formulation of MGTs with MDRs, their quantization, applications in modern astrophysics and cosmology, and search for observable phenomena and experimental verifications.

Nonholonomic frames for Finsler spaces with special (alpha, beta)-metric

The purpose of present paper to determine the Finsler spaces due to deformation of special Finsler $ (\alpha, \beta) $ metric. Consequently, we obtain the non-holonomic frame with help of Riemannian metric $ \alpha $ , one form metric $ \beta $ and Randers metric such as forms I. $L(\alpha, \beta)=\alpha \beta + \beta^{2} $ i.e. product of Randers metric and 1-form metric and II. $L(\alpha, \beta)= \alpha^{2} + \alpha \beta$ i.e. product of Randers metric and Riemannian metric.

Nonholonomic frames for Finsler spaces with special (alpha, beta)-metric

The purpose of present paper to determine the Finsler spaces due to deformation of special Finsler $ (\alpha, \beta) $ metric. Consequently, we obtain the non-holonomic frame with help of Riemannian metric $ \alpha $ , one form metric $ \beta $ and Randers metric such as forms I. $L(\alpha, \beta)=\alpha \beta + \beta^{2} $ i.e. product of Randers metric and 1-form metric and II. $L(\alpha, \beta)= \alpha^{2} + \alpha \beta$ i.e. product of Randers metric and Riemannian metric.

Nonholonomic jet deformations, exact solutions for modified Ricci soliton and Einstein equations

Let [Formula: see text] be a pseudo-Riemannian metric of arbitrary signature on a manifold [Formula: see text] with conventional [Formula: see text]-dimensional splitting, [Formula: see text] determined by a nonholonomic (nonintegrable) distribution [Formula: see text] defining a generalized (nonlinear) connection and associated nonholonomic frame structures. We work with an adapted linear metric compatible connection [Formula: see text] and its nonzero torsion [Formula: see text], both completely determined by [Formula: see text]. Our first goal is to prove that there are certain generalized frame and/or jet transforms and prolongations with [Formula: see text] into explicit classes of solutions of some generalized Einstein equations [Formula: see text], [Formula: see text], encoding various types of (nonholonomic) Ricci soliton configurations and/or jet variables and symmetries. The second goal is to solve additional constraint equations for zero torsion, [Formula: see text], on generalized solutions constructed in explicit forms with jet variables and extract Levi-Civita configurations. This allows us to find generic off-diagonal exact solutions depending on all space time coordinates on [Formula: see text] via generating and integration functions and various classes of constant jet parameters and associated symmetries. Our third goal is to study how such generalized metrics and connections can be related by the so-called “half-conformal” and/or jet deformations of certain subclasses of solutions with one, or two, Killing symmetries. Finally, we present some examples of exact solutions constructed as nonholonomic jet prolongations of the Kerr metrics, with possible Ricci soliton deformations, and characterized by nonholonomic jet structures and generalized connections.

Geometric Framework for Unified Field Theory Using Finsler Gauge Transformation

We study the different types of Finsler space with (α,β)-metrics which have nonholonomic frames as an application for classical mechanics and dynamics in physics using gauge transformation which helps to derive unified field theory. Further, we set up the application of Finsler geometry to geometrize the electromagnetic field completely.

Principal bundle structures among second order frame bundles

Using a model for the bundle Fˆ2M of semi-holonomic second order frames of a manifold M as an extension of the bundle F2M of holonomic second order frames of M, we introduce in Fˆ2M a principal bundle structure over F2M, the structure group being the additive group A2(n) of skew-symmetric bilinear maps from Rn×Rn into Rn. The composition of the projection of that structure with the existing projection of the bundle F˜2M of non-holonomic second order frames of M over Fˆ2M provides a principal bundle structure in F˜2M over F2M. These results close an existing gap in the theory of second order frame bundles.

Off-diagonal ekpyrotic scenarios and equivalence of modified, massive and/or Einstein gravity

Using our anholonomic frame deformation method, we show how generic off-diagonal cosmological solutions depending, in general, on all spacetime coordinates and undergoing a phase of ultra-slow contraction can be constructed in massive gravity. In this paper, there are found and studied new classes of locally anisotropic and (in)homogeneous cosmological metrics with open and closed spatial geometries. The late time acceleration is present due to effective cosmological terms induced by nonlinear off-diagonal interactions and graviton mass. The off-diagonal cosmological metrics and related Stückelberg fields are constructed in explicit form up to nonholonomic frame transforms of the Friedmann–Lamaître–Robertson–Walker (FLRW) coordinates. We show that the solutions include matter, graviton mass and other effective sources modeling nonlinear gravitational and matter fields interactions in modified and/or massive gravity, with polarization of physical constants and deformations of metrics, which may explain certain dark energy and dark matter effects. There are stated and analyzed the conditions when such configurations mimic interesting solutions in general relativity and modifications and recast the general Painlevé–Gullstrand and FLRW metrics. Finally, we elaborate on a reconstruction procedure for a subclass of off-diagonal cosmological solutions which describe cyclic and ekpyrotic universes, with an emphasis on open issues and observable signatures.