Finsler Research Articles

ON THE PROJECTIVE ALGEBRA OF MATSUMOTO SPACE

In the present paper we are study of Matsumoto space on the projective algebra and Lie Algebra of the projective group. The projective Algebra of Matsumoto space is characterized as certain Lie sub algebra of the projective algebra. Further, which is devoted to studying the condition of Finsler space of constant flag curvature and vanishing S curvature admits a non Riemannian space of affine projective vector field with Matsumoto metric is Berwald space.

Born reciprocity and relativistic generalized uncertainty principle in Finsler structure: Fundamental tensor in discretized curved spacetime

The main findings of the generalized uncertainty principle (GUP), the phenomenological approach, for instance, the emergence of a minimal measurable length uncertainty, are obtained in various versions from theories of quantum gravity, such as string theory, loop quantum gravity, doubly special relativity and black hole physics. GUP counts for impacts of relativistic energies and finite gravitational fields on the fundamental theories of quantum mechanics (QM), the noncommutation and measurement uncertainty. Utilizing GUP in reconciling principles of general relativity (GR) and QM, thereby enables to draw convincing conclusions about quantum gravity. To resolve the shortcuts reported with the nonrelativistic three-dimensional GUP, namely, violation of Lorentz covariance, dependence on frame of reference, and violation of the linear additional law of momenta, we introduce relativistic four-dimensional generalized uncertainty principle (RGUP) to curved spacetime. To unify GR and QM, we apply the Born reciprocity principle (BRP), distance-momentum duality symmetry and RGUP to estimate the fundamental tensor in discretized curved spacetime. To this end, we generalize Riemann geometry. The Finsler geometry, which is characterized by manifold and Finsler structure, allows to directly apply RGUP to the Finsler structure of a free particle so that [Formula: see text] can be expressed as [Formula: see text], from which the metric tensor in discretized Riemann spacetime could be deduced. We conclude that [Formula: see text] is homogeneous with degree [Formula: see text] in [Formula: see text], while [Formula: see text] is [Formula: see text]-homogeneous resulting in [Formula: see text]. Despite, the astonishing similarity with the conformal transformation, know as Weyl tensor, this study suggests that principles of QMs could be unambiguously imposed on the resulting fundamental tensor. Also, we conclude that the features of Finsler geometry assumed in this study are likely the ones of the duel Hamilton geometry.

The slope-of-a-mountain problem in a cross gravitational wind

In this paper, the Matsumoto slope-of-a-mountain problem is revisited. We present a general model of the slope as a Riemannian manifold, exploiting the influences of both transverse and longitudinal components of the gravitational wind on time-optimal paths. In contrast to the original Matsumoto’s exposition, the cross-gravity additive is taken into account in the equations of motion, while the along-gravity effect is entirely compensated. The purely geometric solutions to the problem are investigated by means of Finsler geometry. Our study explores the properties of the new cross-slope metric obtained in this work, which belongs to the class of general (α,β)-metrics, emphasizing the strong convexity conditions as well as the corresponding time geodesics. Using some relevant examples the developed theory is illustrated, while explaining and comparing the deformations of the Finslerian indicatrices and the behavior of time-minimal paths on the mountain slopes with different approaches to the gravity effect.

The Characterizations on a Class of Weakly Weighted Einstein–Finsler Metrics

The Characterizations on a Class of Weakly Weighted Einstein–Finsler Metrics

Four-dimensional SO(3)-spherically symmetric Berwald Finsler spaces

We locally classify all [Formula: see text]-invariant four-dimensional pseudo-Finsler Berwald structures. These are Finslerian geometries which are closest to (spatially, or [Formula: see text])-spherically symmetric pseudo-Riemannian ones — and serve as ansatz to find solutions of Finsler gravity equations which generalize the Einstein equations. We find that there exist five classes of non-pseudo-Riemannian (i.e. non-quadratic in the velocities) [Formula: see text]-spherically symmetric pseudo-Finsler Berwald functions, which have either a heavily constrained dependence on the velocities, or, up to a suitable choice of the tangent bundle coordinates, no dependence at all on the “time” and “radial” coordinates.

Automatic tractography and segmentation using finsler geometry based on higher-order tensor fields

Background and objective: We focus on three-dimensional higher-order tensorial (HOT) images using Finsler geometry. In biomedical image analysis, these images are widely used, and they are based on the diffusion profiles inside the voxels. The diffusion information is stored in the so-called diffusion tensor D. Our objective is to present new methods revealing the architecture of neural fibers in presence of crossings and high curvatures. After tracking the fibers, we achieve direct 3D image segmentation to analyse the brain’s white matter structures. Methods: To deal with the construction of the underlying fibers, the inverse of the second-order diffusion tensor D, understood as the metric tensor D−1, is commonly used in DTI modality. For crossing and highly curved fibers, higher order tensors are more relevant, but it is challenging to find an analogue of such an inverse in the HOT case. We employ an innovative approach to metrics based on higher order tensors to track the fibers properly. We propose to feed the tracked fibers as the internal initial contours in an efficient version of 3D segmentation. Results: We propose a brand-new approach to the inversion of a diffusion HOT, and an effective way of fiber tracking in the Finsler setting, based on innovative classification of the individual voxels. Thus, we can handle complex structures with high curvatures and crossings, even in the presence of noise. Based on our novel tractography approach, we also introduce a new segmentation method. We feed the detected fibers as the initial position of the contour surfaces to segment the image using a relevant active contour method (i.e., initiating the segmentation from inside the structures). Conclusions: This is a pilot work, enhancing methods for fiber tracking and segmentation. The implemented algorithms were successfully tested on both synthetic and real data. The new features make our algorithms robust and fast, and they allow distinguishing individual objects in complex structures, even under noise.

On L-reducible spherically symmetric Finsler metrics

In this paper, we study one of the oldest open problems in Finsler geometry which was introduced by Matsumoto-Shimada in 1977 about the existence of a concrete L-reducible Finsler metric that is not C-reducible. To spot such a Finsler metric, we study the class of spherically symmetric Finsler metrics. We prove two rigidity theorems for spherically symmetric Finsler metrics. First, we prove that every spherically symmetric Finsler metric is semi-C-reducible. Second, we show that every non-Riemannian spherically symmetric Finsler metric is a generalized L-reducible metric. Finally, under a particular condition, we prove that every non-Riemannian L-reducible spherically symmetric Finsler metric on a manifold of dimension n≥3 must be a Randers metric. This result provides a negative answer to Matsumoto-Shimada's problem in the class of spherically symmetric Finsler metrics.

On a class of Finsler metrics of quadratic Weyl curvature

The class of warped product manifolds plays an important role in differential geometry and physics. In this paper, we shall study product manifolds $\mathbb{R} \times M$ with Finsler metrics arising from warped product structure. We give an equivalent condition for those metrics to be of quadratic Weyl curvature.

Superconnections and an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds

Superconnections and an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds

Hofer's metric in compact Lie groups

In this article, we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M . Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite-dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.

On Definitions of Finsler Spaces and Axiomatics of Singular Finsler Geometry

A review of various approaches to the concept of a Finsler space based on different definitions is given. In particular, the axiomatics of a singular Finsler space is given.

Radial rapid decay does not imply rapid decay

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group $\Gamma := \mathbf{SL}_2(A)$, where $A := \mathbb{F}_q[X,X^{-1}]$ is the ring of Laurent polynomials with coefficients in $\mathbb{F}_q$, endowed with the length function coming from a natural action of $\Gamma$ on a product of two trees, to show that is has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. The criterion also applies to irreducible lattices in semisimple Lie groups with finite center endowed with a length function defined with the help of a Finsler metric. These examples answer a question asked by Chatterji and moreover show that, unlike the RD property, the RRD property isn't inherited by open subgroups.

Reversible Finsler metrics of constant flag curvature

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric F=F(x,y) on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature K=−1,0,1 respectively. Especially, for the case when K=−1, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when K=1, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.

A Schur type lemma for the mean Berwald curvature in Finsler geometry

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.

On Minkowskian product of Finsler manifolds

Let (M1,F1) and (M2,F2) be a pair of Finsler manifolds. The Minkowskian product Finsler manifold (M,F) of (M1,F1) and (M2,F2) with respect to a product function f is the product manifold M=M1×M2 endowed with the Finsler metric F=f(K,H), where K=F12,H=F22. In this paper, the Cartan connection and Berwald connection of (M,F) are derived in terms of the corresponding objects of (M1,F1) and (M2,F2). Necessary and sufficient conditions for (M,F) to be Berwald (resp. weakly Berwald, Landsberg, weakly Landsberg) manifold are obtained. Thus an effective method for constructing special Finsler manifolds mentioned above is given.

Minimal area of Finsler disks with minimizing geodesics

We show that the Holmes–Thompson area of every Finsler disk of radius r whose interior geodesics are length-minimizing is at least \frac{6}{\pi} r^2 . Furthermore, we construct examples showing that the inequality is sharp and observe that equality is attained by a non-rotationally-symmetric metric. This contrasts with Berger’s conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we include a proof of the integral geometry formulas of Blaschke and Santaló for Finsler manifolds with almost no trapped geodesics.

Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds

‎In this paper‎, ‎we use Hessian comparison and volume comparison theorems to investigate the Mckean-type estimate theorem for the first eigenvalue of p-Laplacian and (p,q)-Laplacian operators on Finsler manifolds‎.

Trajectories of astroparticles in pseudo-Finsler spacetime with the most general modified dispersion

Finsler geometry is a natural and fundamental generalization of Riemann geometry, and is a tool to research Lorentz invariance violation. We find the connection between the most general modified dispersion relation and a pseudo-Finsler structure, and then we calculate the arrival time delay of astroparticles with different modified dispersion relations in the framework of Finsler geometry. The result suggests that the time delay is irrelevant with the exact form of the modified dispersion relation. If the modified term becomes 0 when E=p, there is no arrival time difference, otherwise the time delays only depend on the Lorentz violation scale and the order at which the Lorentz invariance breaks.

Lorentz Violation in Finsler Geometry

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.

On a Projective Class of Finsler Metrics by Warped Product

On a Projective Class of Finsler Metrics by Warped Product