Finsler Research Articles

On the local systolic optimality of Zoll contact forms

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.

On Sprays of Scalar Curvature and Metrizability

Every Finsler metric naturally induces a spray but not so for the converse. The notion for sprays of scalar (resp. isotropic) curvature has been known as a generalization for Finsler metrics of scalar (resp. isotropic) flag curvature. In this paper, a new notion, sprays of constant curvature, is introduced and especially it shows that a spray of isotropic curvature is not necessarily of constant curvature even in dimension \(n\ge 3\). Further, complete conditions are given for sprays of isotropic (resp. constant) curvature to be Finsler metrizable. Based on this result, the local structure is determined for locally projectively flat Berwald sprays of isotropic (resp. constant) curvature which are Finsler metrizable, and some more sprays of isotropic curvature are discussed for their metrizability. Besides, the metrizability problem is also investigated for sprays of scalar curvature under certain curvature conditions.

Transnormal Functions and Focal Varieties on Finsler Manifolds

In this paper, we study transnormal functions and their level sets and focal varieties on complete Finsler manifolds. We prove that the focal varieties of a $$C^{2}$$ transnormal function are smooth submanifolds and each regular level set is a tube over either of the focal varieties.

Gradient estimates of a nonlinear Lichnerowicz equation under the Finsler-geometric flow

Let (Mn, F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow , where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we study gradient estimates for positive solutions of a nonlinear Lichnerowicz equation under Finsler-geometric flow , where c, p are two constants and p > 0. We obtain a global estimate and a Harnack estimate for positive solutions. Our results are also natural extension of similar results on Riemannian-geometric flow.

On the metrizability ofm-Kropina spaces with closed null one-form

We investigate the local metrizability of Finsler spaces with m-Kropina metric F = α1+mβ−m, where β is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric α and one-form β have a very specific form in certain coordinates. In particular, when the signature of α is Lorentzian, α belongs to a certain subclass of the Kundt class and β generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an m-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. In particular, we construct all counterexamples of this type to Szabo’s metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.

Reversibility at infinity and rigidity of Finsler manifolds without conjugate points

Reversibility at infinity and rigidity of Finsler manifolds without conjugate points

Multimetric Finsler geometry

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.

On Study Ch-Trirecurrent Finsler Space

The concept of C^h- recurrent Finsler space has been studied by M.Matsumoto [6] . H.Izumi ([4],[5]) gave the concept of *P- spaces which was the generalization of C^h-recurrent spaces and P2-like spaces of M.Matsumoto ([6],[7]). R.Verma [15] discussed C^h- birecurrent spaces where these spaces are generalization of C^h- recurrent spaces of M.Matsumoto [6] .Besides the correlation of C^h- birecurrent spaces which C^h- recurrent space , some special C^h- birecurrent spaces has been discussed . The result concerning h- isotropic C^h- recurrent space due to M.Matsumoto [6] has been extended to C^h- birecurrent spaces by P.N. Pandey and R.Verma [15]. C.K.Mishra and G.Lodhi [9] studied the properties of C^h- recurrent and C^v- recurrent for torsion tensor field of the second order in Finsler spaces . 
 The purpose of the present paper is to study the properties of C^h- trirecurrent torsion tensor field and the recurrence covariant vector field of the third order in Finsler spaces .

Two short closed geodesics on a sphere of odd dimension

We show that for an open and dense set of non-reversible Finsler metrics on a sphere S^n of odd dimension n=2m-1ge 3 there is a second closed geodesic with Morse index le 4(m+2)(m-1)+2.

On Einstein-reversible mth root Finsler metrics

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root [Formula: see text]-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root [Formula: see text]-metric be Einstein-reversible.

Multiplicity of closed geodesics on bumpy Finsler manifolds with elliptic closed geodesics

Let M be a compact simply connected manifold satisfying H⁎(M;Q)≅Td,n+1(x) for integers d≥2 and n≥1. If all prime closed geodesics on (M,F) with an irreversible bumpy Finsler metric F are elliptic, then either there exist exactly dn(n+1)2 (when d≥2 is even) or (d+1) (when d≥3 is odd) distinct closed geodesics, or there exist infinitely many distinct closed geodesics.

ON Rh -TRIRECURRENT FINSLER SPACES

The concept of recurrent curvature of an n-dimensional Riemannian space was extended to a Finsler space by A. Moór ([7], [8],[9]). Shalini Dikshit [3] defined the birecurrent Finsler space and F.Y.A.Qasem [10] defined the generalized birecurrent Finsler space and their properties considering the Cartan's curvature tensor i  jkh R x , y , y x& .The object of the present paper is to study trirecurrent Finsler space considering the Cartan's curvature tensor ikh R x , y .

Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries

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.

A note on “On the classification of Landsberg spherically symmetric Finsler metrics”

In this paper, we prove that all spherically symmetric Landsberg surfaces are Berwaldian. We modify the classification of spherically symmetric Finsler metrics, done by the author in [S. G. Elgendi, On the classification of Landsberg spherically symmetric Finsler metrics, Int. J. Geom. Methods Mod. Phys. 18 (2021)], of Berwald type of dimension [Formula: see text]. Precisely, we show that all Berwald spherically symmetric metrics of dimension [Formula: see text] are Riemannian or given by a certain formula. As a simple class of Berwaldian metrics, we prove that all spherically symmetric metrics in which the function [Formula: see text] is homogeneous of degree [Formula: see text] in [Formula: see text] and [Formula: see text] are Berwaldian.

Traversable Finslerian wormholes supported by phantom energy

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 <−1, the corresponding exotic matter is none other than the phantom energy. Further, the two solutions in Case ii) are not asymptotically flat since limr→∞[b(r)/r]↛0. To make it asymptotically flat we matched it with the Schwarzschild vacuum.

On the Gromov-Hausdorff theory in Finsler geometry

This paper is devoted to the introduction of the Gromov-Hausdorff theory in Finsler geometry. Firstly, we are able to find a special kind of irreversible metric spaces, named as forward metric spaces, which can accurately describe forward complete Finsler manifolds. Secondly, we establish/generalize the Gromov-Hausdorff theory on such metric spaces. In particular, Gromov's precompactness 定理 remains valid in this setting. Last but not least, we obtain a series of precompactness theorems and stable results in the framework of Finsler geometry. Besides, many interesting examples are presented in this paper, which deeply point out genuine differences between Finsler geometry and Riemannian geometry.

On a class of weakly Landsberg metrics composed by a Riemannian metric and a conformal 1-form

<abstract><p>Without the quadratic restriction, there are many non-Riemannian geometric quantities in Finsler geometry. Among these geometric quantities, Berwald curvature, Landsberg curvature and mean Landsberg curvature are related directly to the famous "unicorn problem" in Finsler geometry. In this paper, Finsler metrics with vanishing weakly Landsberg curvature (i.e., weakly Landsberg metrics) are studied. For the general $ (\alpha, \beta) $-metrics, which are composed by a Riemannian metric $ \alpha $ and a 1-form $ \beta $, we found that if the expression of the metric function doesn't depend on the dimension $ n $, then any weakly Landsberg $ (\alpha, \beta) $-metric with a conformal 1-form must be a Landsberg metric. In the two-dimensional case, the weakly Landsberg case is equivalent to the Landsberg case. Further, we classified two-dimensional Berwald general $ (\alpha, \beta) $-metrics with a conformal 1-form.</p></abstract>

Randers metrics on two-spheres of revolution with simple cut locus

<abstract><p>In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown.</p></abstract>

On projectively flat Finsler space with a cubic (α, β) metric

In the present paper, we have considered a cubic (?, ?) metric which is an special class of p-power Finsler metric, and obtained the conditions under which the Finsler space with such special metric will be projectively flat. Further, we also obtain in which case this Finsler space will be a Berwald space and Douglas space.

Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces

Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces