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In this paper, we introduce a new class of Finsler metrics that generalize the well-known (α, β)-metrics. These metrics are defined by a Riemannian metric α and two 1-forms β = bi(x)yi and γ = γi (x)yi. This new class of metrics not only generalizes (α, β)-metrics, but also includes other important Finsler metrics, such as all (generalized) γ-changes of generalized (α, β)-metrics, (α, β)-metrics, and spherically symmetric Finsler metrics in Rn. We find a necessary and sufficient condition for this new class of metrics to be locally projectively flat. Furthermore, we prove the conditions under which these metrics are of Douglas type. [ABSTRACT FROM AUTHOR]

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for "local instability" in the quantum theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the holonomy group of n-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to D i f f o (S n - 1) , the connected component of the identity of the group of smooth diffeomorphism on the n - 1 -dimensional sphere. In particular, the holonomy group of the n-dimensional standard Funk metric and the Bryant–Shen metrics are maximal and isomorphic to D i f f o (S n - 1) . These results are the firsts describing explicitly the holonomy group of n-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the conditions for mapping the geometric description of the kinematics of particles that probe a given Hamiltonian in phase space to a description in terms of Finsler geometry (and vice-versa). [ABSTRACT FROM AUTHOR]

In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and Prasad (Algebr Geom Topol 23(9):4185–4233, 2023). Given a submanifold N, we consider an N-geodesic loop as an N-geodesic starting and ending in N, possibly at different points. This class of geodesics were studied by Omori (J Differ Geom 2:233–252, 1968). We obtain a generalization of Klingenberg’s lemma for closed geodesics (Klingenberg in: Ann Math 2(69):654–666, 1959). for N-geodesic loops in the reversible Finsler setting. [ABSTRACT FROM AUTHOR]

This study shows, by means of numerical analysis, that the characteristics of discrete dynamical systems, in which chaos and catastrophe coexist, are closely related to the geometric statistics in Finsler geometry. The two geometric statistics introduced are nonlinear connections information, denoted as N I , and the mean deviation curvature, denoted as P ¯. The quantity N I can be used to determine the occurrence of chaos in terms of nonequilibrium stability. The resulting chaos is characterized by P ¯ in terms of the trajectory's robustness, which is related to the localization or globalization of chaos. The characteristics of catastrophe-induced chaos are clearly visualized through the contour topography of N I , in which an abrupt change is represented by cliff topography (i.e. a line of critical points); initial dependence is reflected in the reversibility of topographic patterns. On overlaying the contour topography with the singularity pattern, it is evident that chaos does not arise around the singular point. Furthermore, the extensive development of cusp and butterfly chaos demands information on the nonlinear connections within the singularity pattern. The asymmetry in swallowtail chaos is less distinguishable in an equilibrated state, but becomes more evident when the system is in a state of nonequilibrium. In many analyses, chaos and catastrophe are examined separately. However, these results demonstrate that when both are present, the two have a complex relationship constrained by the singularity. [ABSTRACT FROM AUTHOR]

We construct a unified framework of geometrodynamics based on the Finsler geometry to reveal the relationship between spacetime and dynamics. The Lagrangian of electron in electromagnetic field as the Finsler function gives the Finslerian metric, which modifies spacetime metric in the Finsler-Randers space. The geodesic equation gives the effective mass, forces, and effective (or geometric) fields. Using the Chern connection, we construct the generalized Einstein-Maxwell equations. In the local natural basis, we give generalized Maxwell equations and wave equations. We find that the geometric field couples with electromagnetic field and gives effective charges and currents. We analyze several typical cases, such as flat spacetime, vacuum and Berwald structure. We find that the electromagnetic field coupled with the spacetime torsion emerges in the Berwald space. These results provide some hints to understand some puzzles, such as axion and dark energy. These formulations stimulate some clues to explore deeper geometric structures behind physical phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we find a necessary and sufficient condition for a non-zero vector to be a geodesic vector in homogeneous generalized m-Kropina space. Further, we prove the existence of at least one homogeneous geodesic. However, it is conjectured that the outcomes and proofs in the case of Finsler geometry are not ideal, since general Finsler metrics are non-reversible. In Finsler geometry, the trajectory of unique homogeneous geodesic should be regarded as two geodesics with initial vectors X and -X. Hence, we construct an (n + 1)-dimensional and a 4-dimensional space to find homogeneous geodesics explicitly. [ABSTRACT FROM AUTHOR]

This research aims to shed light on how the quintessential dark energy influences the shadows cast by black holes by considering different kinds of surrounding accretion profiles within the framework of Finsler-Hayward geometry. We explore the impact of the Finsler parameter (η) and the quintessence parameter (ω) on fundamental properties of the black hole, such as its horizons, photon sphere, and impact parameter. By examining the observed characteristics of the black hole's shadow, including the photon ring and shadow intensities, we analyze how the quintessence affects these features under varying scenarios of matter accretion. Leveraging observational data from the Event Horizon Telescope, exclusively focusing on celestial bodies such as Sgr A* and M87*, we narrow down the range of the Finsler parameter (η) within our model of quintessence-infused Finsler Hayward black holes. By comparing our findings with models such as Finsler Schwarzschild, with and without the quintessence Finsler Hayward black holes, we observe that the quintessence's influence on the black hole's shadow surpasses the effects attributed to regularity. This study contributes to our insight into how the Finsler parameter and the quintessence of dark energy shape the shadows of black holes in diverse accretion environments, shedding light on their intricate nature and behavior, particularly in the context of Finsler Hayward geometry.

Two-dimensional time-optimal paths of objects moving under the influence of an external force are discussed based on an analysis of Jacobi stability in Finsler space. When the external force on an object can be described by a function of only one variable, the deviation curvature tensor that determines the Jacobi stability of the object's path can be obtained from the equation of the path. In such cases, the Jacobi stability of the path is represented by the trace of the deviation curvature tensor. The relationship between the Jacobi stability and the type of path is considered for a force that is described by a single-variable trigonometric function. This type of periodic external force induces a path that extends radially and a path along in a specific direction. Then, we consider the time-averaged eigenvalues of the deviation curvature tensor for each type. A large peak in these average values is observed when the type of path changes. Therefore, the Jacobi instability becomes very large at the boundaries between the path types, and the Jacobi stability analysis can be used as the basis of a classification of the path types. [ABSTRACT FROM AUTHOR]

Elasto-plastic deformations of micropolar continuum are discussed by a non-Riemannian geometry. The non-locality of micropolar continuum is described in a second-order vector bundle of displacements and microrotations. With a decomposition of total elasto-plastic field, geometric quantities are divided into the elastic and plastic components independently. Especially, when an intrinsic parallelism of displacements and microrotations holds, integrability conditions of the elasto-plastic field are represented by a torsion tensor or the curvature of nonlinear connection. Then, Burgers and Frank vectors and an energy release rate around crack tips are related to the torsion tensor or the curvature of nonlinear connection. Moreover, the non-locality of microrotation is discussed based on a kink band as a disclination. It is found a generalized expression of Burgers vector which can describe the kink interface including the disclination. [ABSTRACT FROM AUTHOR]

In this article, we study the form of the deviation of geodesics (tidal forces) and the Raychaudhuri equation in a Schwarzschild–Finsler–Randers (SFR) spacetime which has been investigated in previous papers. This model is obtained by considering the structure of a Lorentz tangent bundle of spacetime and, in particular, the kind of the curvatures in generalized metric spaces where there is more than one curvature tensor, such as Finsler-like spacetimes. In these cases, the concept of the Raychaudhuri equation is extended with extra terms and degrees of freedom from the dependence on internal variables such as the velocity or an anisotropic vector field. Additionally, we investigate some consequences of the weak-field limit on the spacetime under consideration and study the Newtonian limit equations which include a generalization of the Poisson equation. [ABSTRACT FROM AUTHOR]

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us, in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position. The analysis appears to provide an underlying geometry described by a geodesic equation, with a connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for “local instability” in the quantum theory.

A continuum mechanical theory with foundations in generalized Finsler geometry describes the complex anisotropic behavior of skin. A fiber bundle approach, encompassing total spaces with assigned linear and nonlinear connections, geometrically characterizes evolving configurations of a deformable body with the microstructure. An internal state vector is introduced on each configuration, describing subscale physics. A generalized Finsler metric depends on the position and the state vector, where the latter dependence allows for both the direction (i.e., as in Finsler geometry) and magnitude. Equilibrium equations are derived using a variational method, extending concepts of finite-strain hyperelasticity coupled to phase-field mechanics to generalized Finsler space. For application to skin tearing, state vector components represent microscopic damage processes (e.g., fiber rearrangements and ruptures) in different directions with respect to intrinsic orientations (e.g., parallel or perpendicular to Langer's lines). Nonlinear potentials, motivated from soft-tissue mechanics and phase-field fracture theories, are assigned with orthotropic material symmetry pertinent to properties of skin. Governing equations are derived for one- and two-dimensional base manifolds. Analytical solutions capture experimental force-stretch data, toughness, and observations on evolving microstructure, in a more geometrically and physically descriptive way than prior phenomenological models. [ABSTRACT FROM AUTHOR]

In the standard approach to Finsler geometry the metric is defined as a vertical Hessian and the Chern or Cartan connections appear as just two among many possible natural linear connections on the pullback tangent bundle. Here it is shown that the Hessian nature of the metric, the nonlinear connection and the Chern or Cartan connections can be derived from a few compatibility axioms between metric and Finsler connection. This result provides a metrical formulation of Finsler geometry which is well adapted to field theory, and which has proved useful in Einstein–Cartan-like approaches to Finsler gravity. [ABSTRACT FROM AUTHOR]

We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of Anosov subgroups given in our earlier work, extending the class from intrinsically hyperbolic to relatively hyperbolic subgroups. We prove implications and equivalences between the various relativizations. [ABSTRACT FROM AUTHOR]

In this paper, we reviewtwo approaches that can describe, in a geometrical way, the kinematics of particles that are affected by Planck-scale departures, named Finsler and Hamilton geometries. By relying on maps that connect the spaces of velocities and momenta, we discuss the properties of configuration and phase spaces induced by these two distinct geometries. In particular, we exemplify this approach by considering the so-called q-de Sitter-inspired modified dispersion relation as a laboratory for this study. We finalize with some points that we consider as positive and negative ones of each approach for the description of quantum configuration and phases spaces.

In this article, we study the form of the deviation of geodesics (tidal forces) and the Raychaudhuri equation in a Schwarzschild–Finsler–Randers (SFR) spacetime which has been investigated in previous papers. This model is obtained by considering the structure of a Lorentz tangent bundle of spacetime and, in particular, the kind of the curvatures in generalized metric spaces where there is more than one curvature tensor, such as Finsler-like spacetimes. In these cases, the concept of the Raychaudhuri equation is extended with extra terms and degrees of freedom from the dependence on internal variables such as the velocity or an anisotropic vector field. Additionally, we investigate some consequences of the weak-field limit on the spacetime under consideration and study the Newtonian limit equations which include a generalization of the Poisson equation.

A Finsler analog of the Lobachevsky plane is the Lie group of proper affine transformations of the real line with a left-invariant Finsler structure generated by a convex compact set in the Lie algebra with the origin in its interior. We consider the isoperimetric problem on this Lie group, with the volume form also taken to be left-invariant. This problem is formulated as an optimal control problem. Applying the Pontryagin maximum principle, we find the optimal isoperimetric loops in an explicit form in terms of convex trigonometry functions. We also present a generalized isoperimetric inequality in a parametric form. [ABSTRACT FROM AUTHOR]

Lorentz invariance is one of the foundations of modern physics; however, Lorentz violation may happen from the perspective of quantum gravity, and plenty of studies on Lorentz violation have arisen in recent years. As a good tool to explore Lorentz violation, Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. Here, we simply introduce the mathematics of Finsler geometry. We review the connection between modified dispersion relations and Finsler geometries and discuss the physical influence from Finsler geometry. We review the connection between Finsler geometries and theories of Lorentz violation, such as the doubly special relativity, the standard-model extension, and the very special relativity. [ABSTRACT FROM AUTHOR]

For the general class of pseudo-Finsler spaces with (α , β) -metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has a Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future-pointing time-like vectors is ensured. The identified (α , β) -Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an (α , β) -metric and the isometries of the underlying pseudo-Riemannian metric a; in particular, we list all (α , β) -metrics which admit isometries that are not isometries of a. [ABSTRACT FROM AUTHOR]

The reversibility of asymptotic classes of geodesics in the universal covering of a Riemannian visibility manifold is a consequence of the reversibility of geodesics. The purpose of this article is to study reversibility at infinity for Finsler visibility manifolds and its relationship with geodesic reversibility. We show that a compact Finsler metric without conjugate points and visibility universal covering is p-reversible at infinity for every p in a fundamental domain if and only if the Finsler metric is geodesically reversible, namely, geodesics with reversed parametrization remain geodesics of the Finsler metric. We get as well some rigidity results for magnetic flows in compact surfaces. Let (M, g) be a compact Riemannian surface without focal points and with genus greater than one. We show that if the g-geodesic curvature of the magnetic geodesics has constant sign and the magnetic flow has no conjugate points, then p-reversibility at infinity for p in an open set in the universal covering of the surface implies that the magnetic flow is trivial. Under the same assumption on the g-geodesic curvature of magnetic geodesics, assuming that the magnetic flow is expansive we show that p-reversibility at just one point in the universal covering implies that the magnetic flow is trivial. [ABSTRACT FROM AUTHOR]

We consider a geometrical approach to the optimisation problems motivated by transportation system management. Here, we provide a comprehensive account of geometric programming based on the elementary Finsler geometry in R n . Then, we present a Finslerian dynamical model for the nonlinear complementarity problem of traffic equilibrium that can be applied to a variety of equilibrium problems. [ABSTRACT FROM AUTHOR]

After a brief introduction of Born's reciprocal relativity theory is presented, we review the construction of the d e f o r m e d quaplectic group that is given by the semi-direct product of U (1 , 3) with the d e f o r m e d (noncommutative) Weyl–Heisenberg group corresponding to n o n c o m m u t a t i v e fiber coordinates and momenta [ X a , X b ] ≠ 0 ; [ P a , P b ] ≠ 0 . This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [ X a 1 a 2 ⋯ a n , X b 1 b 2 ⋯ b n ] ≠ 0 ; [ P a 1 a 2 ⋯ a n , P b 1 b 2 ⋯ b n ] ≠ 0 . We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting x μ , p μ operator variables (associated to an 8D curved phase space) to the canonical Y A , Π A operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the c l a s s i c a l limit, the embedding functions Y A (x , p) , Π A (x , p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric g μ ν (x , p) , the fiber metric of the vertical space h a b (x , p) , and the nonlinear connection N a μ (x , p) associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we reviewtwo approaches that can describe, in a geometrical way, the kinematics of particles that are affected by Planck-scale departures, named Finsler and Hamilton geometries. By relying on maps that connect the spaces of velocities and momenta, we discuss the properties of configuration and phase spaces induced by these two distinct geometries. In particular, we exemplify this approach by considering the so-called q-de Sitter-inspired modified dispersion relation as a laboratory for this study. We finalize with some points that we consider as positive and negative ones of each approach for the description of quantum configuration and phases spaces. [ABSTRACT FROM AUTHOR]

This note is devoted to present the scientific work of Professor Csaba Varga (1959-2021), who had contributions in Calculus of Variations and its applications in the theory of Partial Differential Equations and Finsler Geometry. [ABSTRACT FROM AUTHOR]

Motivated in part by the bi-gravity approach to massive gravity, we introduce and study the multimetric Finsler geometry. For the case of an arbitrary number of dimensions, we study some general properties of the geometry in terms of its Riemannian ingredients, while in the two-dimensional case, we derive all the Cartan equations as well as explicitly find the Holmes–Thompson measure. [ABSTRACT FROM AUTHOR]

Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are multiclass classification and class probability estimation. Central to their definition is the choice of loss function, which is the means by which the quality of a solution is evaluated. In this paper we systematically develop the theory of loss functions for such problems from a novel perspective whose basic ingredients are convex sets with a particular structure. The loss function is defined as the subgradient of the support function of the convex set. It is consequently automatically proper (calibrated for probability estimation). This perspective provides three novel opportunities. It enables the development of a fundamental relationship between losses and (anti)-norms that appears to have not been noticed before. Second, it enables the development of a calculus of losses induced by the calculus of convex sets which allows the interpolation between different losses, and thus is a potential useful design tool for tailoring losses to particular problems. In doing this we build upon, and considerably extend, existing results on M-sums of convex sets. Third, the perspective leads to a natural theory of "polar" loss functions, which are derived from the polar dual of the convex set defining the loss, and which form a natural universal substitution function for Vovk's aggregating algorithm. [ABSTRACT FROM AUTHOR]

With the help of a generalization of the Fermat principle in generalrelativity, we show that chains in CR geometry are geodesics of a certainKropina metric constructed from the CR structure. We study the projectiveequivalence of Kropina metrics and show that if the kernel distributions of thecorresponding 1-forms are non-integrable then two projectively equivalentmetrics are trivially projectively equivalent. As an application, we show thatsufficiently many chains determine the CR structure up to conjugacy,generalizing and reproving the main result of [J.-H. Cheng, 1988]. Thecorrespondence between geodesics of the Kropina metric and chains allows us touse the methods of metric geometry and the calculus of variations to studychains. We use these methods to re-prove the result of [H. Jacobowitz, 1985]that locally any two points of a strictly pseudoconvex CR manifolds can bejoined by a chain. Finally, we generalize this result to the global setting byshowing that any two points of a connected compact strictly pseudoconvex CRmanifold which admits a pseudo-Einstein contact form with positiveTanaka-Webster scalar curvature can be joined by a chain.

This paper supplements the article (Eur. Phys. J. C 76: 246, 2016) by extending some more new wormhole solutions in the background of Finslerian geometry. Here, we present six more solutions by considering i) a linear equation of state (EoS) pr = ωρ and three different shape functions, ii) a linear relationship between radial and tangential pressure as pt = npr with two different redshift functions and iii) a specific density profile with a linear equation of state pr = ωρ. It is found that all these wormholes are violating null energy condition (NEC) signifying that the throats are opened by exotic matters. Since the equation of state parameters, ω or ωr = pr/ρ for all the solutions are

In this work, we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and the manifold of strictly accretive unitary matrices. Utilizing this decomposition, we introduce a family of Finsler metrics on the manifold and characterize their geodesics and geodesic distances. Finally, we apply the geodesic distance to a matrix approximation problem and also give some comments on the relation between the introduced geometry and the geometric mean of strictly accretive matrices as defined by Drury [1]. [ABSTRACT FROM AUTHOR]

We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a (3+1)-dimensional perspective these can be viewed as dispersion relations for waves propagating in the string interior and are analogous to those for current-carrying topological defects. This allows us to construct a unified framework with which to study and interpret the internal structure of various field-theoretic and fundamental string species, in a simple physically intuitive coordinate system, without the need for dimensional reduction or approximate effective actions. This, in turn, allows us to identify the precise conditions under which higher-dimensional strings and current-carrying defects are observationally indistinguishable, for macroscopic observers. Our approach naturally incorporates the description of so-called 'cosmic springs', whose dynamics are expressed in terms of an effective Finsler geometry, for circular loops, or generalised Finsler geometry, for non-circular configurations. This demonstrates the importance of these novel geometric structures and their utility in modelling complex physical phenomena in cosmology and astrophysics. [ABSTRACT FROM AUTHOR]

This research aims to shed light on how the quintessential dark energy influences the shadows cast by black holes by considering different kinds of surrounding accretion profiles within the framework of Finsler-Hayward geometry. We explore the impact of the Finsler parameter (η) and the quintessence parameter (ω) on fundamental properties of the black hole, such as its horizons, photon sphere, and impact parameter. By examining the observed characteristics of the black hole's shadow, including the photon ring and shadow intensities, we analyze how the quintessence affects these features under varying scenarios of matter accretion. Leveraging observational data from the Event Horizon Telescope, exclusively focusing on celestial bodies such as Sgr A* and M87*, we narrow down the range of the Finsler parameter (η) within our model of quintessence-infused Finsler Hayward black holes. By comparing our findings with models such as Finsler Schwarzschild, with and without the quintessence Finsler Hayward black holes, we observe that the quintessence's influence on the black hole's shadow surpasses the effects attributed to regularity. This study contributes to our insight into how the Finsler parameter and the quintessence of dark energy shape the shadows of black holes in diverse accretion environments, shedding light on their intricate nature and behavior, particularly in the context of Finsler Hayward geometry. [ABSTRACT FROM AUTHOR]

A continuum mechanical theory with foundations in generalized Finsler geometry describes the complex anisotropic behavior of skin. A fiber bundle approach, encompassing total spaces with assigned linear and nonlinear connections, geometrically characterizes evolving configurations of a deformable body with the microstructure. An internal state vector is introduced on each configuration, describing subscale physics. A generalized Finsler metric depends on the position and the state vector, where the latter dependence allows for both the direction (i.e., as in Finsler geometry) and magnitude. Equilibrium equations are derived using a variational method, extending concepts of finite-strain hyperelasticity coupled to phase-field mechanics to generalized Finsler space. For application to skin tearing, state vector components represent microscopic damage processes (e.g., fiber rearrangements and ruptures) in different directions with respect to intrinsic orientations (e.g., parallel or perpendicular to Langer’s lines). Nonlinear potentials, motivated from soft-tissue mechanics and phase-field fracture theories, are assigned with orthotropic material symmetry pertinent to properties of skin. Governing equations are derived for one- and two-dimensional base manifolds. Analytical solutions capture experimental force-stretch data, toughness, and observations on evolving microstructure, in a more geometrically and physically descriptive way than prior phenomenological models.

Using Monte Carlo (MC) simulations, we study the skyrmion stability/instability as a response to uniaxial mechanical stresses. Skyrmions emerge in chiral magnetic materials as a stable spin configuration under external magnetic field B→with the competition of ferromagnetic interaction and Dzyaloshinskii–Moriya interaction (DMI) at low temperature T. Skyrmion configurations are also known to be stable (unstable) under a compressive stress applied parallel (perpendicular) to B→. To understand the origin of such experimentally confirmed stability/instability, we use the Finsler geometry modeling technique with a new degree of freedom for strains, which plays an essential role in DMI being anisotropic. We find from MC data that the area of the skyrmion state on the B−Tphase diagram increases (decreases) depending on the direction of applied stresses, in agreement with reported experimental results. This change in the area of the skyrmion state indicates that skyrmions become more (less) stable if the tensile strain direction is parallel (perpendicular) to B→. From the numerical data in this paper, we find that the so-called magneto-elastic effect is suitably implemented in the effective DMI theory with the strain degree of freedom without complex magneto-elastic coupling terms for chiral magnetic materials. This result confirms that experimentally-observed skyrmion stability and instability are caused by DMI anisotropy.

Lorentz invariance is one of the foundations of modern physics; however, Lorentz violation may happen from the perspective of quantum gravity, and plenty of studies on Lorentz violation have arisen in recent years. As a good tool to explore Lorentz violation, Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. Here, we simply introduce the mathematics of Finsler geometry. We review the connection between modified dispersion relations and Finsler geometries and discuss the physical influence from Finsler geometry. We review the connection between Finsler geometries and theories of Lorentz violation, such as the doubly special relativity, the standard-model extension, and the very special relativity.

For the general class of pseudo-Finsler spaces with (α,β)-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has a Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future-pointing time-like vectors is ensured. The identified (α,β)-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an (α,β)-metric and the isometries of the underlying pseudo-Riemannian metric a; in particular, we list all (α,β)-metrics which admit isometries that are not isometries of a.

In this paper, we demonstrate that Robb-Geroch's definition of a relativistic interval admits a simple and fairly natural generalization leading to a Finsler extension of special relativity. Another justification for such an extension goes back to the works of Lalan and Alway and, finally, was put on a solid basis and systematically investigated by Bogoslovsky under the name "Special-relativistic theory of locally anisotropic spacetime". The isometry group of this spacetime, DISIM b (2) , is a deformation of the Cohen and Glashow's very special relativity symmetry group ISIM(2). Thus, the deformation parameter b can be regarded as an analog of the cosmological constant characterizing the deformation of the Poincaré group into the de Sitter (anti-de Sitter) group. The simplicity and naturalness of Finslerian extension in the context of this paper adds weight to the argument that the possibility of a nonzero value of b should be carefully considered. [ABSTRACT FROM AUTHOR]

Finsler differential geometry enables enriched mathematical and physical descriptions of the mechanics of materials with microstructure. The first propositions for Finsler geometry in solid mechanics emerged some six decades ago. Ideas set forth in these early works are reviewed, along with subsequent literature culminating in contemporary theories of Finsler-geometric continuum mechanics. Concepts unique to generalized Finsler spaces, in the context of continuum mechanical applications, are highlighted. Capabilities afforded by physical models in generalized Finsler spaces are contrasted with those of standard approaches in affinely connected spaces. Theory and several examples of reduced dimensionality are reported for boundary value problems of fracture and phase transformations, showing how simultaneously novel, physical, and pragmatic model predictions can be obtained from Finsler-type continuum field theory. Lastly, the modern theory is newly applied to describe nonlinear elastic ferromagnetic solids in the magnetically saturated state. A variational approach is used to derive Euler–Lagrange equations for macroscopic and microscopic, i.e., respective electromechanical and electronic continuum, equilibrium states. For a representative generalized Finsler metric depending on material symmetry, augmented conservation laws of macroscopic momentum and electronic spin angular momentum naturally emerge. [ABSTRACT FROM AUTHOR]

We consider a natural mechanical system on a Finsler manifold and study its curvature using the intrinsic Jacobi equations (called Jacobi curves) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (involving the gradient of the potential) of the Finsler manifold and the Hessian of the potential w.r.t. a Riemannian metric induced from the Finsler metric. As an application, we give sufficient conditions for the Hamiltonian flows of the least action to be hyperbolic and show new examples of Anosov flows. [ABSTRACT FROM AUTHOR]

After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the deformed quaplectic group that is given by the semi-direct product of U(1,3) with the deformed (noncommutative) Weyl–Heisenberg group corresponding to noncommutative fiber coordinates and momenta [Xa,Xb]≠0; [Pa,Pb]≠0. This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [Xa1a2⋯an,Xb1b2⋯bn]≠0; [Pa1a2⋯an,Pb1b2⋯bn]≠0. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting xμ,pμ operator variables (associated to an 8D curved phase space) to the canonical YA,ΠA operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, the embedding functions YA(x,p),ΠA(x,p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric gμν(x,p), the fiber metric of the vertical space hab(x,p), and the nonlinear connection Naμ(x,p) associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions.

Determining and modeling the wind speed characteristics of a region are important in terms of constructing the wind energy conversion systems. Several distribution functions such as Finsler geometry, two-parameter Weibull, and Rayleigh are proposed for wind speed modeling in the literature. The modeling performance of Finsler geometry method at high and low speeds was not investigated in the literature, although a model proposal was presented in the studies on Finsler geometry. In addition, there is no comparison in terms of power density. This paper presents the comparative performance analysis of Finsler geometry for modeling the wind speed data. The Finsler geometry method allows accurate modeling and describes the ability for chaotic structures like wind speed data. The two-parameter Weibull, Rayleigh, and Finsler Geometry are used to analyze the wind speed data between October 2015 and September 2016 in Bilecik, Gökçeada, and Bozcaada, which are located in the northwest of Turkey. The obtained results show that the novel method based on Finsler geometry is a better alternative to the two-parameter Weibull and the Rayleigh probability density functions to describe wind speed characteristics. [ABSTRACT FROM AUTHOR]

We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a (3+1)-dimensional perspective these can be viewed as dispersion relations for waves propagating in the string interior and are analogous to those for current-carrying topological defects. This allows us to construct a unified framework with which to study and interpret the internal structure of various field-theoretic and fundamental string species, in a simple physically intuitive coordinate system, without the need for dimensional reduction or approximate effective actions. This, in turn, allows us to identify the precise conditions under which higher-dimensional strings and current-carrying defects are observationally indistinguishable, for macroscopic observers. Our approach naturally incorporates the description of so-called ‘cosmic springs’, whose dynamics are expressed in terms of an effective Finsler geometry, for circular loops, or generalised Finsler geometry, for non-circular configurations. This demonstrates the importance of these novel geometric structures and their utility in modelling complex physical phenomena in cosmology and astrophysics.