Finsler Research Articles

Sharp Morrey–Sobolev inequalities and eigenvalue problems on Riemannian–Finsler manifolds with nonnegative Ricci curvature

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey–Sobolev inequalities on [Formula: see text]-dimensional Finsler manifolds having nonnegative [Formula: see text]-Ricci curvature. A byproduct of this method is a Hardy–Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler–Laplace operator. Our results are also new in the Riemannian setting.

Further Geometric and Lyapunov Characterizations of Incrementally Stable Systems on Finsler Manifolds

In this paper, we report several new geometric and Lyapunov characterizations of incrementally stable systems on Finsler and Riemannian manifolds. A new and intrinsic proof of an important theorem in contraction analysis is given via the complete lift of the system. Based on this, two Lyapunov characterizations of incrementally stable systems are derived, namely, converse contraction theorems, and revelation of the connection between incremental stability and stability of an equilibrium point, in which the second result recovers and extends the classical Krasovskii's method.

Sasakian structure on the unit tangent bundle of a Finsler manifold

In this work, we introduce an adopted local frame on the tangent bundle of a Finsler manifold with respect to the natural foliations of the tangent bundle. We show the prominence of using this local frame by studying some geometric properties of the foliations and distributions on the tangent bundle of a Finsler manifold. Moreover, we find the necessary and sufficient conditions on the Finsler manifold ( M, F ) such that the unit tangent bundle admits a Sasakian structure.

Busemann Functions in Asymptotically Harmonic Finsler Manifolds

In the present paper, we investigate Busemann functions in a general Finsler setting as well as in asymptotically harmonic Finsler manifolds. In particular, we show Busemann functions are smooth on asymptotically harmonic Finsler manifolds.

Patterson–Sullivan theory for Anosov subgroups

We extend several notions and results from the classical Patterson–Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.

Finsler metrics and semi-symmetric compatible linear connections

Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds Wagner (Dokl Acad Sci USSR (N.S.) 39:3–5, 1943). Compatible linear connections are the solutions of the so-called compatibility equations containing the components of the torsion tensor as unknown quantities. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics Bartelmeß and Matveev (J Diff Geom Appl 58:264–271, 2018), extremal compatible linear connections and algorithmic solutions Vincze (Aequat Math 96:53–70, 2022)), it is very hard to solve them in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved Vincze (Publ Math Debrecen 83(4):741–755, 2013 (see also Vincze (Euro J Math 3:1098–1171, 2017))) that such a compatible linear connection must be uniquely determined. The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the uniqueness by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities.

First integrals for Finsler metrics with vanishing $$\chi $$-curvature

We prove that in a Finsler manifold with vanishing \(\chi \)-curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature.

Two-body decays in deformed relativity

Deformed relativistic kinematics is a framework which captures effects, that are expected from particles and fields propagating on a quantum spacetime, effectively. They are formulated in terms of a modified dispersion relation and a modified momentum conservation equation. In this work we use Finsler geometry to formulate deformed relativistic kinematics in terms of particle velocities. The relation between the Finsler geometric velocity dependent formulation and the original momentum dependent formulation allows us to construct deformed Lorentz transformations between arbitrary frames. Moreover, we find the corresponding compatible momentum conservation equation to first order in the Planck scale deformation of special relativity based on the κ-Poincaré algebra in the bicrossproduct basis. We find that the deformed Lorentz transformations, as well as the deformed time dilation factor, contain terms that scale with the energy of the particle under consideration to the fourth power. We derive how the distributions of decay products are affected when the deformed relativity principle is satisfied and find, for the case of a pion decaying into a neutrino and a muon, that the ratio of expected neutrinos to muons with a certain energy is just slightly modified when compared to the predictions based on special relativity. We also discuss the phenomenological consequences of this framework for cosmic-ray showers in the atmosphere.

SOME GEOMETRICAL RESULTS ON NEARLY KÄHLER FINSLER MANIFOLDS

This work is intended as an attempt to extend some results of nearly Kählerian Finsler manifolds. We give a condition to generalized $ (a, b, {\bf J})- $manifolds to be weakly Landsberg metric. Furthermore, we find the conditions under which a nearly Kähler Finsler manifold has relatively isotropic Landsberg curvature and relatively isotropic mean Landsberg curvature.

Isoparametric hypersurfaces in conic Finsler manifolds

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in conic Finsler spaces. We find that there are probably other isoparametric hypersurfaces in conic Minkowski spaces besides the conic Minkowski hyperplanes, conic Minkowski hyperspheres and conic Minkowski cylinders, such as helicoids. Moreover, we give a complete classification of isoparametric hypersurfaces in Kropina spaces with constant flag curvature.

The second closed geodesic, the fundamental group, and generic Finsler metrics

For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also extend results about generic Riemannian metrics to Finsler metrics. We show a bumpy metrics theorem for Finsler metrics and prove that a C^4-generic Finsler metric on a compact and simply-connected manifold carries infinitely many closed geodesics.

Recurrent curvature tensors in Finsler spaces with generalised connection

Recurrent curvature tensors in Finsler spaces with generalised connection

On projective invariants based on non-linear connections in a Finsler space

On projective invariants based on non-linear connections in a Finsler space

Projective recurrent Finsler manifolds. I.

Projective recurrent Finsler manifolds. I.

Generalisation of Gauss--Codazzi equations for Berwald's curvature tensor in a hypersurface of a Finsler space

Generalisation of Gauss--Codazzi equations for Berwald's curvature tensor in a hypersurface of a Finsler space

On some properties of Finsler spaces based on the indicatrices

On some properties of Finsler spaces based on the indicatrices

Super geodesic congruence in a subspace of a Finsler space

Super geodesic congruence in a subspace of a Finsler space

Lorentz-violation-induced arrival time delay of astroparticles in Finsler spacetime

Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. We present the arrival time delay of astroparticles subject to Lorentz violation in the framework of Finsler geometry, and the result corresponds to that derived by Jacob and Piran in the standard model of cosmology.

Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

We establish sharp universal upper bounds on the length of the shortest closed geodesic on a punctured sphere with three or four ends endowed with a complete Riemannian metric of finite area. These sharp curvature-free upper bounds are expressed in terms of the area of the punctured sphere. In both cases, we describe the extremal metrics, which are modeled on the Calabi–Croke sphere or the tetrahedral sphere. We also extend these optimal inequalities for reversible and non-necessarily reversible Finsler metrics. In this setting, we obtain optimal bounds for spheres with a larger number of punctures. Finally, we present a roughly asymptotically optimal upper bound on the length of the shortest closed geodesic for spheres/surfaces with a large number of punctures in terms of the area.

A novel family of edge preserving anisotropic filters

In this article, new anisotropic image filters are introduced and their performances are compared with existing isotropic and anisotropic filters. The developed filters were examined according to their image noise removing performances in terms of standard image quality metrics, as well as the edge preserving properties of the filtered images. Mathematical inferences of anisotropic filters are made based on the minimization of the Polyakov action energy integral. New anisotropic metrics are found by means of Finsler metrics that minimize the corresponding integral. The new metrics perform filtering by updating the image with anisotropic Laplace-Beltrami flow. After filtering, it was observed that the introduced metrics perform well against other anisotropic metrics. It is also observed that the developed New Randers, New Normalized Miron, and New Metric filters preserve edges better than other filters, making it a plausible noise removal tool prior to edge detection in image processing. The source codes of proposed filters are publicly available at https://github.com/HAYDARKILIC.