Finsler Research Articles

On generalized recurrent Finsler space of fifth order in Berwald sense

On generalized recurrent Finsler space of fifth order in Berwald sense

On some non-Riemannian curvature of Minkowskian product Finsler metrics

Let (M1,F1) and (M2,F2) be two Finsler manifolds, Minkowskian product Finsler metric is the Finsler metric F=f(S,T) endowed on the product manifold M=M1×M2, where S=F12, T=F22, f is product function. In this paper, the stretch curvature, mean stretch curvature, χ-curvature and H-curvature of F are derived in terms of the corresponding objects of its components. It turns out that F is a stretch metric or weakly stretch metric if and only if F1 and F2 both are stretch metric or weakly stretch metric. Necessary and sufficient conditions for F to be of vanishing χ-curvature or H-curvature are obtained. Under certain condition, we obtain a differential equation to characterize F with almost vanishing χ-curvature. We also prove that F with almost vanishing H-curvature has vanishing H-curvature.

Mechanics of geodesics in information geometry and black hole thermodynamics

In this paper, we shall discuss the theory of geodesics in information geometry, and an application in astrophysics. We will study how gradient flows in information geometry describe geodesics, explore the related mechanics by introducing a constraint, and apply our theory to Gaussian model and black hole thermodynamics. Thus, we demonstrate how deformation of gradient flows leads to more general Randers–Finsler metrics, describe Hamiltonian mechanics that derive from a constraint, and prove duality via canonical transformation. We also verified our theories for a deformation of the Gaussian model, and described dynamical evolution of flat metrics for Kerr and Reissner–Nordström black holes.

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

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.

Positively curved Finsler metrics on vector bundles, II

Positively curved Finsler metrics on vector bundles, II

On the Pontrjagin classes of spray manifolds

Abstract The characterization of projectively flat Finsler metrics on an open subset in $R^n$ is the Hilbert’s fourth problem in the regular case. Locally projectively flat Finsler manifolds form an important class of Finsler manifolds. Every Finsler metric induces a spray on the manifold via geodesics. Therefore, it is a natural problem to investigate the geometric and topological properties of manifolds equipped with a spray. In this paper, we study the Pontrjagin classes of a manifold equipped with a locally projectively flat spray and show that such manifold must have zero Pontrjagin classes.

On conformal transformations preserving the Ricci tensor in Finsler geometry

Here we obtain a classical integral formula on the conformal change of Finsler metrics. As an application, we obtain significant results depending on the sign of the Ricci scalars, for mean Landsberg surfaces and show there is no conformal transformation between two compact mean Landsberg surfaces, one of a non-positive Ricci scalar and another of a non-negative Ricci scalar, except for the case where both Ricci scalars are identically zero. Conformal transformations preserving the Ricci tensor are known as Liouville transformations. Here we show that a Liouville transformation between two compact mean Landsberg manifolds of isotropic S-curvature is homothetic. Moreover, every Liouville transformation between two compact Finsler n-manifolds of bounded mean value Cartan tensor is homothetic. These results are an extension of the results of M. Obata and S. T. Yau on Riemannian geometry and give a positive answer to a conjecture on Liouville's theorem.

Study on generalized BK-5th recurrent Finsler space

In this paper, we present a novel new class and investigate the connection between the K-projective curvature tensor and other tensors of Finsler space Fn, this space is characterized by the property for Cartan’s 4th curvature tensor satisfies the certain relationship with the given covariant vectors field, we define this space as a generalized BK-5th recurrent space and denote it briefly by GBK-5RFn. This paper aims to derive the fifth-order Berwald covariant derivatives of the torsion tensor and the deviation tensor . Additionally, it demonstrates that the curvature vector Kj, the curvature vector Hk, and the curvature scalar H are all non-vanishing within the considered space. We have identified tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal fifth-order Berwald covariant derivatives with their lower-order counterparts.

Finsler Space of Fourth Order on Some Properties of Rh-Generalized Recurrent

In the present paper‚ a Finsler space whose Cartan’s fourth curvature tensor R_jkh^i satisfies R_(jkh|l|m|n|s)^i=c_(lmns R_jkh^i+d_lmns δ_h^i g_jk) , R_jkh^i ≠ 0 , where ׀l׀m|n|s is h-covariant derivative of fourth order ( Cartan’s third kind covariant differential operator ), with respect to x^l , x^m , x^n and x^s successively, c_(lmns ) and d_(lmns ) are non-zero covariant vector field and covariant tensor field of third order, respectively‚ is introduced and such space is called as R^h -generalized fourecurrent Finsler space and we denote them briefly by R^h-G-FR f_n‚ we obtained some generalized fourecurrent space. Also we introduced Ricci generalized fourecurrent space.

Warped f-product Finsler Metrics

This paper shows the existence of some Ricci-flat warped f-Product Finsler metrics. We investigate the general structure of this newly defined warped f-Product Finsler metrics, indeed we identify the metric form, spray coefficients of geodesics of the metric, and also the Ricci curvature in regards to the α_1 and α_2 Riemanian metrics.

On pseudo-Riemannian quartics in Finsler geometry

Finsler geometry usually describes an extension of Riemannian geometry into a direction-dependent geometric structure. Historically, the well-known Riemann quartic length element example served as the inspiration for this construction. Surprisingly, the same quartic expression emerges as a fundamental dispersion relationcovariant Fresnel equationin solid-state electrodynamics. As a result, it is possible to conceive of the Riemann quartic length expression as a mathematical representation of a well-known physical phenomenon. This paper provides a number of Riemann quartic examples that show Finsler geometry to be overly constrictive for many applications, even when the signature space is positive definite in the Euclidean sense. The strong axioms of Finsler geometry are broken down on many more singular hypersurfaces for the spaces having an indefinite (Minkowski) signature. We suggest a more flexible definition of a Finsler structure that only has to hold for open subsets of a manifold’s tangent bundle. We demonstrate the distinctive singular hypersurfaces connected to the Riemann quartic and discuss the potential physics explanations for them. As an illustration of the pseudo-Riemannian quartic, we took into consideration the dispersion relation that appears in electromagnetic wave propagation in uniaxial crystal. Our analysis suggests that the signature of the Finsler measure may be altered for large anisotropy factors.

On the non-Riemannian quantity Ξ-curvature of (α,β)-metrics

The non-Riemannian quantity Ξ-curvature which is very useful to characterize the Finsler metrics with constant flag curvature forms an important class in Finsler geometry. In this paper, we classify the (α,β)-metrics with almost isotropic Ξ-curvature.

Conformal gravitational theories in Barthel–Kropina-type Finslerian geometry, and their cosmological implications

We consider dark energy models obtained from the general conformal transformation of the Kropina metric, representing an (alpha , beta )-type Finslerian geometry, constructed as the ratio of the square of a Riemannian metric alpha and the one-form beta . Conformal symmetries appear in many fields of physics, and they may play a fundamental role in our understanding of the Universe. We investigate the possibility of obtaining conformal theories of gravity in the osculating Barthel–Kropina geometric framework, where gravitation is described by an extended Finslerian-type model, with the metric tensor depending on both the base space coordinates and a vector field. We show that it is possible to formulate a family of conformal Barthel–Kropina theories in an osculating geometry with second-order field equations, depending on the properties of the conformal factor, whose presence leads to the appearance of an effective scalar field of geometric origin in the gravitational field equations. The cosmological implications of the theory are investigated in detail by assuming a specific relation between the component of the one-form of the Kropina metric and the conformal factor. The cosmological evolution is thus determined by the initial conditions of the scalar field and a free parameter of the model. We analyze in detail three cosmological models corresponding to different values of the theory parameters. Our results show that the conformal Barthel–Kropina model can provide an acceptable description of the observational data, and may represent a theoretically attractive alternative to the standard Lambda CDM cosmology.

Volume comparison theorems in Finsler geometry

Volume comparison theorems in Finsler geometry

Homeomorphisms between Finsler manifolds satisfying the Rohde condition

Homeomorphisms between Finsler manifolds satisfying the Rohde condition

A study of recurrent Finsler spaces of higher order with Cartan’s Curvature Tensor

In the present communication, we have derived Bianchi and Veblen identities along with a few more related results in a recurrent and generalized nth-recurrent Finsler space with Cartan’s curvature tensor field. A Finsler space \(F_n\) whose Cartan's third curvature tensor \(R_{jkh}^i\) satisfies the condition \(R_{jkh|m_1|m_2|…|m_n}^{i} = λ_{m_1m_2…m_n} R_{jkh}^{i} + μ_{m_1m_2…m_n} (δ_{h}^{i} g_{jk} - δ_{k}^{i} g_{jh})\), where \(R_{jkh}^i\)≠0 and \(|m_1 |m_2 |…|m_n\) are h-covariant differentiation (Cartan's second kind covariant differential operator) with respect to x^m to nth order, \(λ_{m_1 m_2…m_n }\) and \(μ_{m_1 m_2…m_n }\) is recurence tensors fields.

Homeomorphisms between Finsler manifolds satisfying the Rohde condition

We study the main features of homeomorphisms between two Finsler manifolds satisfying the Rohde condition, i.e. the difference between the moduli of geodesic rings in the image and its preimage is uniformly bounded from below and above. Some equivalent conditions for such a characterization are established. Also we investigate the absolute continuity and the relations to the class of mappings with finite are distortion. As applications, the boundary correspondence results for homeomorphisms satisfying the Rohde condition between regular domains are also presented.

FINSLER METRICS WITH BUSEMANN CURVATURE BOUNDS

We prove that a Finsler metric has Busemann curvature bounded above (below, respectively) by $\kappa$ if and only if it is the Berwald metric with flag curvature bounded above (below, respectively) by $\kappa$. Combining this with Szabó’s Berwald metrization theorem, we can obtain that such a Finsler metric is affinely equivalent to a Riemannian metric with sectional curvature bounded above (below, respectively) by $\kappa$.

A novel finite element approximation of anisotropic curve shortening flow

We extend the DeTurck trick from the classical isotropic curve shortening flow to the anisotropic setting. Here, the anisotropic energy density is allowed to depend on space, which allows an interpretation in the context of Finsler metrics, giving rise to, for instance, geodesic curvature flow in Riemannian manifolds. Assuming that the density is strictly convex and smooth, we introduce a novel weak formulation for anisotropic curve shortening flow. We then derive an optimal H^1 -error bound for a continuous-in-time semidiscrete finite element approximation that uses piecewise linear elements. In addition, we consider some fully practical fully discrete schemes and prove their unconditional stability. Finally, we present several numerical simulations, including some convergence experiments that confirm the derived error bound, as well as applications to crystalline curvature flow and geodesic curvature flow.

A class of critical surfaces in a Finsler space under the volume preserving variation

A class of critical surfaces in a Finsler space under the volume preserving variation