On "Regular Landsberg metrics are always Berwald" by Z. I. Szabo

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 $n\geq 3$. Precisely, we show that all Berwald spherically symmetric metrics of dimension $n\geq 3$ 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 funct…

On the classifiation of Landsberg spherically symmetric Finsler metrics

In this paper, as an application of the inverse problem of calculus of variations, we investigate two compatibility conditions on the spherically symmetric Finsler metrics. By making use of these conditions, we focus our attention on the Landsberg spherically symmetric Finsler metrics. We classify all spherically symmetric manifolds of Landsberg or Berwald types. For the higher dimensions $n\geq 3$, we prove that: all Landsberg spherically symmetric manifolds are either Riemannian or their geodesic sprays have a specific formula; all regular Landsberg spherically symmetric metrics are Riemanni…

Solutions for the Landsberg unicorn problem 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 explicit examples of non-Berwaldian Landsberg metrics. In fact, such kinds of examples are very tedious and complicated to investigate. Nonetheless, we use the maple program and Finsler packages to simplify the calculations in an elegant way. Depending on these examples, we manage …

Special left invariant conic Finsler metrics and homogeneous conic Landsberg Problem in two dimension

In this paper, we study left invariant conic Finsler metrics on the 2-dimensional non-Abelian Lie group $G$ with nowhere vanishing spray vector fields, and classify those satisfying the constant curvature condition, the Landsberg condition or the Berwald condition respectively. We prove that any left invariant conic Landsberg metric on $G$ must be Berwald. This discovery enable us to propose a homogeneous conic Landsberg Conjecture, which guesses that every homogeneous conic Landsberg metric is Berwald, and prove the 2-dimensional case for it.

关于vanishing Landsberg曲率的广义Finsler扭曲乘积度量

On a generalized Finsler warped product metrics with vanishing Landsberg curvature

本文研究弱正交不变Finsler度量，推导其Berwald和Landsberg曲率的显式表达式，并推导出广义Finsler扭曲乘积度量的偏微分方程系统，应用于构造非正则Landsberg度量的例子。

Properties of Berwald scalar curvature

In this short paper, we prove that a Finsler manifold with vanishing Berwald scalar curvature has zero $\mathbf{E}$-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. This improves a previous result in \cite{Li}. For $(α,β)$-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, the Berwald curvature also vanishes.

Doubly Warped Product Finsler Manifolds with Some Non-Riemannian Curvature Properties

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 if and only it is Berwaldian. In continue, we prove that every relatively isotropic Landsberg DWP-manifold is a Landsberg manifold. We show that a relatively isotropic mean Landsberg warped product manifold is a weakly Landsberg manifold. Finally, we show that there is no any lo…

There are no $\mathcal{C}^5$-Regular Pure $y$-Global Landsberg Surfaces

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 cases and showing that in dimension 2 Landsberg condition implies Berwald condition.

On Special Berwald Metrics

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 metrics, the notion of Landsberg and weakly Landsberg curvature are equivalent.

On Landsberg general $(α,β)$-metrics with a conformal 1-form

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 prove that regular Landsberg general $(α,β)$-metrics must be Berwaldian when the dimension is greater than two. For the almost regular case, the classification also is given and some new non-Berwaldian Landsberg metrics are found.

闭三维正则Landsberg度量的刚性

The Rigidity of the closed three dimensional regular Landsberg metrics

该研究证明闭三维流形上所有光滑强凸正则Landsberg度量均为Berwald，解决了Landsberg-Berwald猜想，通过结合消没定理、刚性定理和秩一论证实现。

An observation on Asanov's Unicorn metrics

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 - global) Landsberg metrics that are not of Berwald type is an open problem up to this day. In this paper we prove that Asanov's Unicorns belong to the class of generalized Berwald manifolds. More precisely we prove the following theorems: a Finsleroid-Finsler space is a generaliz…

On general $(α, β)$-metrics of weak Landsberg type

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 $(α,β)$-metric. Further, We show that $F$ is a Landsberg general $(α,β)$-metric if and only if it is weak Landsberg general $(α,β)$-metric, where $β$ is a closed and conformal one-form.

On cubic Berwald spaces

We show that, for Finsler spaces with cubic metric, Landsberg spaces are Berwaldian. Also, for decomposable metrics, we determine specific conditions for a space with cubic metric to be of Berwald type, thus refining the result in [6].

On Cartan Spaces with m-th Root Metrics

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 Landsberg and weakly Berwald space, respectively. Then we show that m-th root Cartan space of almost vanishing H-curvature satisfies H = 0.

宇宙学Landsberg Finsler时空

Cosmological Landsberg Finsler spacetimes

本文在4维中分类了宇宙学均匀各向同性的Landsberg型Finsler结构，识别出非平稳时空，并构造了FLRW几何的唯一非Berwald Landsberg推广。

General $(α, β)$ metrics with relatively isotroic mean Landsberg curvature

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 mean Landsberg curvature to be Berwald.

A new look at Finsler surfaces and the Landsberg's PDE

In this paper, we introduce a new look at Finsler surfaces. Landsberg surfaces are Finsler surfaces that are solutions of a system of non-linear partial differential equations. Considering the unicorn's Landsberg problem, we reduce this system to a single non-linear PDE which we call the Landsberg's PDE. By making use of the new look of Finsler surfaces, we solve the Landsberg's PDE and get a class of solutions. Moreover, we show that these solutions and their conformal transformations are Berwaldain.

The Landsberg equation of a Finsler space

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 of the Landsberg type must be a Berwald metric. This shows that the hunting for a unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful even in the very broad class of $(α_1,α_2)$-metrics.

Generalized Isotropic Berwald Manifolds

In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald manifold, we show that the notions of stretch and Landsberg curvatures are equivalent. Then we prove that on these manifolds, a Finsler metric is R-quadratic if and only if it is a stretch metric with vanishing E -curvature. Finally, we determine the flag curvature of generalized is…

Some curvature properties of Cartan spaces with mth root metrics

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 root metric be locally dually flat. Finally, we prove that the conformal $β$-change of locally projectively flat m-th root metrics are locally Minkowskian.

满足σT条件的球对称芬斯勒度量

Spherically symmetric Finsler metrics satisfying the $σ$T-condition

研究满足σT条件的球对称芬斯勒度量，通过刻画该度量及研究其在兰斯贝格类中的子类，确定相关条件，还构造了经典独角兽问题新解，给出非贝尔瓦尔德的兰斯贝格度量例子。

T-张量为零的Finsler曲面

Finsler surfaces with vanishing $T$-tensor

该微分几何研究证明Finsler曲面上T-条件与σT-条件等价，高维下二者不等价，刻画了Berwald、Landsberg曲面非位似共形变换保持类型的充要条件，并完成满足该条件的Finsler曲面分类。

On Minkowskian Product of Finsler Manifolds

Let (M_1,F_1) and (M_2,F_2) be a pair of Finsler manifolds. The Minkowskian product Finsler manifold (M,F) of (M_1,F_1) and (M_2,F_2) with respect to a product function f is the product manifold M=M_1\times M_2 endowed with the Finsler metric F^2=f(K,H), where K=(F_1)^2,H=(F_2)^2. In this paper, the Cartan connection and Berwald connection of (M,F) are derived in terms of the corresponding objects of (M_1,F_1) and (M_2,F_2). Necessary and sufficient conditions for (M,F) to be Berwald (resp. weakly Berwald, Landsberg, weakly Landsberg) manifold are obtained. Thus an effective method for constru…

Some Rigidity Conditions on Berwald Structures

This thesis contains an introduction to the method of average in Finsler geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity conditions. We prove that the Levi-Civita connection of any Riemannian metric affine equivalent to the Berwald metric leaves invariant the indicatrix of th Finsler metric F. A converse result also holds: if (M,F) is a Finsler structure such that there is a Riemannian connection whose Levi-Civita leaves invariant by parallel transport the indicatrix of the Finsler structure, then the structure (M,F) is Berwald. As an application we obtain a nece…

特殊芬斯勒流形上的平行1-形式

Parallel one forms on special Finsler manifolds

本文研究特殊芬斯勒流形上平行1-形式的存在性，推导了Landsberg流形相关曲率秩的结论，证明了几类特殊芬斯勒度量均不容许平行1-形式。

A note on L. Zhou's result on Finsler surfaces with $K = 0$ and $J = 0$

In this note, we show that the examples of non Berwaldian Landsberg surfaces with vanishing flag curvature, obtained in \cite{Zhou}, are in fact Berwaldian. Consequently, Bryant's claim is still unverified.

Riemannian metrics having common geodesics with Berwald metrics

In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F. As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; this statement is a "Berwald" version of Hilbert's 4th problem. Further, we investigate geodesic equivalence of Berwald metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwald metric admits a Riemannian metric that is…

Projective relations for m-th root metric spaces

For Finsler spaces (M,F) endowed with m-th root metrics, we provide necessary and sufficient conditions in which they are projectively flat, or projectively related to Berwald/Riemann spaces. We also give a specific characterization for m-th root metrics spaces of Landsberg and of Berwald type.

关于非零旗帜曲率的贝瓦尔德空间

On Berwald Spaces with non-Zero Flag Curvature

该研究证明非零旗帜曲率的贝瓦尔德流形是黎曼流形，并扩展了刚性定理，同时探讨了各向同性贝瓦尔德流形的性质及推广了贝瓦尔德-兰德斯度量的结果。

广义Berwald射影Weyl度量

Generalized Berwald Projective Weyl metrics

本文在芬斯勒几何中提出广义Berwald射影Weyl度量，证明其C-射影不变性、所属GDW度量类的相关性质，拓展了Numata定理。

Ten ways to Berwald manifolds -- and some steps beyond

After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-free covariant derivative on the base manifold compatible with the Finsler function and Aikou's characterization of Berwald manifolds. Finally, we study some implications of V. Matveev's observation according to which quadratic convexity may be omitted from the definition of a Berwa…

On a Class of Generalized Berwald Manifolds

The class of generalized Berwald metrics contains the class of Berwald metrics. In this paper, we characterize two-dimensional generalized Berwald $(α, β)$-metrics with vanishing S-curvature. Let $F=αϕ(s)$, $s=β/α$, be a two-dimensional generalized Berwald $(α,β)$-metric on a manifold $M$. Suppose that $F$ has vanishing S-curvature. We show that one of the following holds: (i) if $F$ is a regular metric, then it reduces to a Riemannian metric of isotropic sectional curvature or a locally Minkowskian metric; (ii) if $F$ is an almost regular metric that is not Riemannian nor locally Minkowskian,…

On the projective theory of sprays with applications to Finsler geometry

Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at least 3-dimensional Finsler manifold with vanishing projected Berwald tensor is a Berwald manifold. To obtain consequences in two dimensions, we transcript Berwald's classical theory on 2-dimensional Finsler manifolds in our setup. We prove criteria and necessary conditions for Fin…

关于Finsler流形上测地喷雾的F-协变系数

On the covariant coefficients of geodesic sprays on Finsler manifolds

本文引入Finsler度量测地喷雾的F-协变系数与S-标量概念，证明S-标量存在当且仅当度量对偶平坦，并研究射影平坦情形下的线性联络性质，进而提出H-Berwald与H-Landsberg空间并解决相关Landsberg问题。

On Asanov's Finsleroid-Finsler metrics as the solutions of a conformal rigidity problem

Finsleroid-Finsler metrics form an important class of singular (y-local) Finsler metrics. They were introduced by G. S. Asanov [2] in 2006. As the special case of the general construction Asanov produced singular (y - local) examples of Landsberg spaces of dimension at least three that are not of Berwald type. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day; for a detailed exposition of the so-called unicorn problem in Finsler geometry see D. Bao [3]. In this paper we are going to characterize the Finsleroid-Finsler metrics…

The curvatures of spherically symmetric Finsler metrics in $R^n$

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial differential equation systems which can respectively characterize the spherically symmetric Finsler metrics with constant flag curvature and Einstein metrics of this type is also obtained. Utilizing these equations, we find an effective way to construct the non-projective, non-Randers Finsler metrics with constant flag curvature and many explicit examples are given by…

锥伪芬斯勒曲面的各向异性共形变换，II

Anisotropic conformal change of conic pseudo-Finsler surfaces, II

本文研究锥伪芬斯勒曲面的各向异性共形变换，建立 Berwald、Landsberg、Douglas 张量及 T-张量的对应关系，并给出黎曼曲面变换为各类特殊曲面及保持 T-条件的充要条件。

Recent results on complex Cartan spaces

In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan and strongly Berwald-Cartan spaces. Then, we investigate the complex geodesic curves of a complex Cartan space, using the image by Legendre transformation ($\mathcal{L}-$ duality) of complex geodesic curves of a complex Finsler space. Assuming the weakly Kähler condition for a c…

On Non-Positively Curved Homogeneous Finsler Metrics

In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. We prove that a homogeneous $(α, β)$-metrics has i…

The Euler-Lagrange PDE and Finsler metrizability

In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial differential system on the energy function can be reduced to a first order system on this same function. In this way we are able to give effective necessary and sufficient conditions for the local existence of a such Finsler metric in terms of the holonomy algebra generated by horizontal vector-fields. We also consider the Landsberg metrizability problem and prove sim…

Pseudo-Finsler spaces modeled on a pseudo-Minkowski space

We adopt a vierbein formalism to study pseudo-Finsler spaces modeled on a pseudo-Minkowski space. We show that it is possible to obtain closed expressions for most of the geometric objects of the theory, including Berwald's curvature, Landsberg's tensor, Douglas' curvature, non-linear connection and Ricci scalar. These expressions are particularly convenient in computations since they factor the dependence on the base and the fiber. As an illustration, we study Lorentz-Finsler spaces modeled on the Bogoslovsky Lorentz-Minkowski space, and give sufficient conditions which guarantee the Berwald …

Geodesic and curvature of piecewise flat Finsler surfaces

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. Using the edge-crossing equation, we define two classes of piecewise flat Finsler surfaces, namely, Landsberg type and Berwald type. We deduce an explicit condition for a geodesic to be extendable at a vertex, and define the curvature which measures the \textit{amount} of such …

On the non metrizability of Berwald Finsler spacetimes

We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern-Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi-Civita connection of some Riemannian metric. We show for the first time that this result does not extend to Finsler spacetimes. More precisely, we find a large class of Berwald spacet…

First integrals for Finsler metrics with vanishing $χ$-curvature

We prove that in a Finsler manifold with vanishing $χ$-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.

On the metrizability of $m$-Kropina spaces with closed null 1-form

We investigate the local metrizability of Finsler spaces with $m$-Kropina metric $F = α^{1+m}β^{-m}$, where $β$ is a closed null 1-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric $α$ and 1-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 s…