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Keywords = Berwald space

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16 pages, 571 KiB  
Article
KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction
by M. K. Gupta, Abha Sahu, C. K. Yadav, Anjali Goswami and Chetan Swarup
Axioms 2023, 12(12), 1133; https://doi.org/10.3390/axioms12121133 - 18 Dec 2023
Cited by 1 | Viewed by 2688
Abstract
The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system [...] Read more.
The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system of second-order differential equations. In this approach, we describe the evolution of the Oregonator model in geometric terms, by considering it as a geodesic in a Finsler space. We have found five KCC invariants using the general expression of the nonlinear and Berwald connections. To understand the chaotic behavior of the Oregonator model, the deviation vector and its curvature around equilibrium points are studied. We have obtained the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points. Further, a comprehensive examination was conducted to compare the linear stability and Jacobi stability of the Oregonator model at its equilibrium points, and We highlight these instances with a few illustrative examples. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)
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11 pages, 301 KiB  
Article
Curvatures on Homogeneous Generalized Matsumoto Space
by M. K. Gupta, Suman Sharma, Fatemah Mofarreh and Sudhakar Kumar Chaubey
Mathematics 2023, 11(6), 1316; https://doi.org/10.3390/math11061316 - 9 Mar 2023
Cited by 2 | Viewed by 1521
Abstract
The curvature characteristics of particular classes of Finsler spaces, such as homogeneous Finsler spaces, are one of the major issues in Finsler geometry. In this paper, we have obtained the expression for S-curvature in homogeneous Finsler space with a generalized Matsumoto metric [...] Read more.
The curvature characteristics of particular classes of Finsler spaces, such as homogeneous Finsler spaces, are one of the major issues in Finsler geometry. In this paper, we have obtained the expression for S-curvature in homogeneous Finsler space with a generalized Matsumoto metric and demonstrated that the homogeneous generalized Matsumoto space with isotropic S-curvature has to vanish the S-curvature. We have also derived the expression for the mean Berwald curvature by using the formula of S-curvature. Full article
(This article belongs to the Special Issue Geometry of Manifolds and Applications)
23 pages, 1063 KiB  
Article
Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole
by Cristina Blaga, Paul Blaga and Tiberiu Harko
Symmetry 2023, 15(2), 329; https://doi.org/10.3390/sym15020329 - 24 Jan 2023
Cited by 9 | Viewed by 2409
Abstract
We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the [...] Read more.
We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results. Full article
(This article belongs to the Special Issue BGL 2022: Dedicated to the Creators of the Non-euclidean Geometry: János Bolyai (220th Anniversary) and N.I. Lobachevski (230th Anniversary))
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14 pages, 798 KiB  
Article
Cosmological Finsler Spacetimes
by Manuel Hohmann, Christian Pfeifer and Nicoleta Voicu
Universe 2020, 6(5), 65; https://doi.org/10.3390/universe6050065 - 5 May 2020
Cited by 37 | Viewed by 3093
Abstract
Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lemaître-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes [...] Read more.
Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lemaître-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes that can be regarded as closest to pseudo-Riemannian geometry. They are defined by a Finsler Lagrangian built from a zero-homogeneous function on the tangent bundle, which encodes the velocity dependence of the Finsler Lagrangian in a very specific way. The obtained cosmological Berwald geometries are candidates for the description of the geometry of the universe, when they are obtained as solutions from a Finsler gravity equation. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
12 pages, 291 KiB  
Article
On the Non Metrizability of Berwald Finsler Spacetimes
by Andrea Fuster, Sjors Heefer, Christian Pfeifer and Nicoleta Voicu
Universe 2020, 6(5), 64; https://doi.org/10.3390/universe6050064 - 1 May 2020
Cited by 18 | Viewed by 2825
Abstract
We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the [...] Read more.
We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi–Civita connection of some Riemannian metric. We show for the first time that this result does not extend to general Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of a metric-affine geometry with vanishing torsion. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
13 pages, 321 KiB  
Article
On the Analyticity of Static Solutions of a Field Equation in Finsler Gravity
by Erasmo Caponio and Antonio Masiello
Universe 2020, 6(4), 59; https://doi.org/10.3390/universe6040059 - 22 Apr 2020
Cited by 18 | Viewed by 2630
Abstract
It is well-known that static vacuum solutions of Einstein equations are analytic in suitable coordinates. We ask here for an extension of this result in the context of Finsler gravity. We consider Finsler spacetimes that retain several properties of static Lorentzian spacetimes, are [...] Read more.
It is well-known that static vacuum solutions of Einstein equations are analytic in suitable coordinates. We ask here for an extension of this result in the context of Finsler gravity. We consider Finsler spacetimes that retain several properties of static Lorentzian spacetimes, are Berwald and have vanishing Ricci scalar. Full article
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
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