arXiv research

Solves a PDE for Landsberg surfaces using new Finsler surface insights.

Paper proves every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.

Note refutes examples of Landsberg surfaces with vanishing flag curvature.

The paper studies invariant functions and their relation to Landsberg surfaces.

The Berwald-Landsberg problem is considered for two dimensional manifolds. We sketch the proof that there are not $\mathcal{C}^5$-regular $y$-global pure Landsberg surfaces. The method used consists on considerer the holonomy representation of the averaged Chern connection and then exhausting all the possible cases, sh…

We show that there are not pure $\mathcal{C}^5$ regular y-global Landsberg surfaced. The proof is based on the averaged connection associated with the linear Chern's connection and the classification of irreducibles holonomies of torsion-free affine connections. The structure consists on exausting all the possible case…

The paper classifies Finsler surfaces satisfying the T-condition or σT-condition.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…

The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.

This paper addresses the problem of existence of generalized Landsberg structures on surfaces using the Cartan-Kähler Theorem and a Path Geometry approach.

Paper studies Landsberg curvature of a specific Finsler metric.

In this short note, we verify R. Bryant's claim: there does exist the singular Landsberg Finsler surface with a vanishing flag curvature which is not Berwaldian.

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.

The paper explores parallel 1-forms on special Finsler manifolds and their properties.

The paper studies anisotropic conformal changes in pseudo-Finsler surfaces.

The paper provides an intrinsic proof of a theorem about Landsberg spaces.

In this paper, we study the complex Landsberg spaces and some of their important subclasses. We introduce and characterize the class of generalized Berwald and complex Landsberg spaces. The intersection of these spaces gives the so called $G-$ Landsberg class.

The paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.

In this paper, we study the long existence problem of non Berwaldian Landsberg spaces using the conformal transformation point of view. Under conformal transformation, the Berwald and Landesberg tensors are calculated in terms of the T-tensor. By giving examples, we show that under conformal transformation, there are L…

Study Finsler metrics with vanishing Landsberg curvature.

In this paper, we study almost regular Landsberg general $(α,β)$-metrics in Finsler geometry. The corresponding equivalent equations are given. By solving the equations, we give the classification of Landsberg general $(α,β)$-metrics under the conditon that $β$ is closed and conformal to $α$. Under this condition, we p…

Classifies cosmological Finsler spacetimes, finding viable non-stationary models.

We show that which that for a Berwald structure, any Riemannian structure that is preserved by the Berwald connection leaves the indicatrix invariant under horizontal parallel transport. We also obtain the converse result: if $({\bf M},F)$ is a Finsler structure such that there exists a Riemannian structure that leaves…

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider Cartan space with the m-th root metric. We prove that every m-th root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly …

Study on special anisotropic conformal changes of conic pseudo-Finsler surfaces.

The paper studies Berwald scalar curvature properties in Finsler geometry.

It is still a long-standing open problem in Finsler geometry, is there any regular Landsberg metric which is not Berwaldian. However, there are non-regular Landsberg metrics which are not Berwladian. The known examples are established by G. S. Asanov and Z. Shen. In this paper, we use the Maple program to study some ex…

Given a Finsler space, we introduce a system of partial differential equations, called the Landsberg equation. Based on a careful analysis of the Landsberg equation and the observation that the solution space is invariant under the linear isometries of the tangent Minkowski spaces, we prove that an $(α_1, α_2)$-metric …

In this paper, we study general $(α,β)$-metrics which $α$ is a Riemannian metric and $β$ is an one-form. We have proven that every weak Landsberg general $(α,β)$-metric is a Berwald metric, where $β$ is a closed and conformal one-form. This show that there exist no generalized unicorn metric in this class of general $(…

Schur theorem proven for weakly Landsberg Finsler metrics.

In this paper, we consider doubly warped product (DWP) Finsler manifolds with some non-Riemannian curvature properties. First, we study Berwald and isotropic mean Berwald DWP-Finsler manifolds. Then we prove that every proper Douglas DWP-Finsler manifold is Riemannian. We show that a proper DWP-manifold is Landsbergian…

Study left invariant spray structures on Lie groups, calculating curvature and geodesics.

A gap in the proof of the finsler "unicorn" conjecture in the paper "Regular Landsberg metrics are always Berwald" by Z. I. Szabo is pointed out

The problem of conformal transformation and conformal flatness of Finsler spaces has been studied by so many researchers $\left[ 6,16,17,20,21\right] .$ Recently, Prasad et. al $\left[ 19\right]$ have studied three dimensional conformally flat Landsberg and Berwald spaces and have given some important results. The pur…

In this Note, we prove that every m-th root Finsler metric with isotropic Landsberg curvature reduces to a Landsberg metric. Then, we show that every m-th root metric with almost vanishing H-curvature has vanishing H-curvature.

Characterizes spherical Finsler metrics satisfying a specific condition.

Given a Finsler space (M,F), one can define natural average Riemannian metrics on M by averaging on the indicatrix I_x the fundamental tensor g of the Finsler function $F$. In this paper we determine explicitly the Levi-Civita connection for these average Riemannian metrics. We apply the result to the case when (M,F) i…

By performing required evaluations, we show that in the Finsleroid-regular space the Landsberg-space condition just degenerates to the Berwald-space condition (at any dimension number $N\ge2$). Simple and clear expository representations are obtained. Due comparisons with the Finsleroid-Finsler space are indicated. Key…

We formulate the notion of the Finsleroid--Finsler space, including the positive--definite as well as indefinite cases. The associated concepts of angle, scalar product, and the distance function are elucidated. If the Finsleroid--Finsler space is of Landsberg type, then the Finsleroid charge is a constant. The Finsler…

In this paper, we study a class of Finsler metrics which contains the class of Berwald metrics as a special case. We prove that every Finsler metric in this class is a generalized Douglas-Weyl metric. Then we study isotropic flag curvature Finsler metrics in this class. Finally we show that on this class of Finsler met…

Study new warped metrics with vanishing Douglas curvature.

The paper proves conjectures about Minkowski norms with specific symmetry groups.

In this paper, we study a new class of Finsler metrics, F=αφ(b^2,s), s:=β/α, defined by a Riemannian metric αand 1-form β. It is called general (α, β) metric. In this paper, we assume φbe coefficient by s and βbe closed and conformal. We find a nessecary and sufficient condition for the metric of relatively isotropic m…

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y -…

Finsler gravity vacuum equation reduces to Ricci vanishing under specific conditions.

Paper studies Minkowskian product of Finsler manifolds and their connections.

In this paper, we prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal $β$-change of an m-th…