The paper examines (α,β)-metrics and proves they are Berwald metrics under certain conditions.
problem Characterizing (α,β)-metrics of weak Landsberg type. method Analyzing properties of (α,β)-metrics, proving conditions for weak Landsberg metrics to be Berwald. result Every weak Landsberg (α,β)-metric is a Berwald metric when β is closed and conformal. Paper studies Landsberg curvature of a specific Finsler metric.
problem Analyzing Landsberg curvature of a particular Finsler metric.
method Derived Landsberg curvature and mean Landsberg curvature of the twisted product Finsler metric.
result Necessary and sufficient conditions for the metric to be Landsberg or weakly Landsberg are established.
Study on non-Berwaldian Landsberg spaces using conformal transformation.
problem Existence and properties of non-Berwaldian Landsberg spaces.
method Using conformal transformation to analyze Berwald and Landsberg tensors.
result Necessary condition for a Landsberg space to be Berwaldian is given.
Paper proves every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
problem Characterizing homogeneous Landsberg surfaces.
method Proved isotropic flag curvature and used it to prove rigidity.
result Every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.
The paper classifies Landsberg metrics under specific conditions.
problem Classifying Landsberg metrics with conformal 1-form.
method Solving equivalent equations for Landsberg metrics.
result Regular Landsberg metrics are Berwaldian under certain conditions.
The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.
problem Investigating compatibility conditions on spherically symmetric Finsler metrics.
method Using the inverse problem of calculus of variations, the paper focuses on Landsberg and Berwald types.
result All Landsberg spherically symmetric manifolds in higher dimensions are either Riemannian or have specific geodesic spray formulas.
Solves a PDE for Landsberg surfaces using new Finsler surface insights.
problem Solving the Landsberg's PDE for Finsler surfaces.
method Reduces the system of non-linear PDEs to a single PDE, the Landsberg's PDE, and solves it.
result Obtains a class of solutions for the Landsberg's PDE.
The paper provides an intrinsic proof of a theorem about Landsberg spaces.
problem Proving Numata's theorem on Landsberg spaces of scalar curvature.
method Intrinsic point of view and coordinate-free proof using Finsler geometry.
result All Landsberg spaces of dimension n≥3 of non-zero scalar curvature are Riemannian spaces of constant curvature. 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.
problem Classifying Landsberg metrics on a 2D Lie group.
method Analyzing left invariant conic Finsler metrics on a 2D non-Abelian Lie group.
result Any left invariant conic Landsberg metric on G must be Berwald. Study Finsler metrics with vanishing Landsberg curvature.
problem Characterize Finsler metrics with specific curvature properties.
method Derive expressions for curvatures, solve PDEs, construct examples.
result Construct non-regular Landsberg metrics not of Berwald type.
The Berwald-Landsberg problem is considered for two dimensional manifolds. We sketch the proof that there are not C5-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…
New Landsberg metrics found without being Berwaldian.
problem Existence of regular Landsberg metrics that are not Berwaldian.
method Used Maple program to investigate explicit examples and deformed spray methods.
result Found new non-Berwaldian Landsberg metrics and a formula.
Classifies cosmological Finsler spacetimes, finding viable non-stationary models.
problem Locating viable non-stationary Finsler spacetimes in cosmology.
method Locally classified all possible cosmological homogeneous and isotropic Landsberg-type Finsler structures in 4-dimensions.
result Identified unique Finsler, non-Berwaldian Landsberg generalization of Friedmann-Lemaitre-Robertson-Walker geometry.
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 …
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
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 (α1,α2)-metric …
Schur theorem proven for weakly Landsberg Finsler metrics.
problem Proving the Schur theorem for a specific class of Finsler metrics.
method Using the Ricci curvature and properties of the mean Landsberg tensor, the theorem is proven for weakly Landsberg metrics.
result For weakly Landsberg Finsler metrics, the Ricci scalar must be constant.
Note refutes examples of Landsberg surfaces with vanishing flag curvature.
problem Verifying examples of Landsberg surfaces with specific curvature properties.
method Analyzing examples from Zhou's result to show they are Berwaldian.
result Examples of Landsberg surfaces with vanishing flag curvature are Berwaldian.
Note on special curvature properties of m-th root metrics.
problem Characterizing m-th root metrics with specific curvature properties.
method Analyzing isotropic Landsberg curvature and almost vanishing H-curvature.
result m-th root metrics with isotropic Landsberg curvature reduce to Landsberg 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…
The paper studies invariant functions and their relation to Landsberg surfaces.
problem Investigating the geometry of invariant functions and their applications to Landsberg surfaces.
method Investigating the geometry of S-invariant functions and their associated vertical subdistribution, and relating the holonomy distribution to these subdistributions. result For Landsberg surfaces, if the flag curvature is S-invariant, it is constant, and the surface is Riemannian. The paper extends the study of conformally flat spaces to four dimensions.
problem Understanding conformal transformations and flatness in four-dimensional Finsler spaces.
method Extending the study of three-dimensional conformally flat Landsberg and Berwald spaces to four dimensions.
result Conditions for a four-dimensional conformally flat Landsberg space to become a Berwald space.
We show that there are not pure C5 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…
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 paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.
problem Classifying spherically symmetric Finsler metrics.
method Proving all spherically symmetric Landsberg surfaces are Berwaldian and modifying the classification of spherically symmetric Finsler metrics.
result All Berwald spherically symmetric metrics of dimension n≥3 are Riemannian or given by a specific formula. Paper studies new Finsler metrics with specific curvature properties.
problem Investigates new Finsler metrics with isotropic mean Landsberg curvature.
method Defines and analyzes general (α, β) metrics under specific conditions.
result Identifies necessary and sufficient conditions for relatively isotropic mean Landsberg curvature.
Characterizes spherical Finsler metrics satisfying a specific condition.
problem Spherically symmetric Finsler metrics satisfying the σT-condition. method Complete characterization and subclass investigation within Landsberg category.
result Identifies precise conditions for metrics satisfying the T-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≥2). Simple and clear expository representations are obtained. Due comparisons with the Finsleroid-Finsler space are indicated. Key…
The paper explores curvature properties of specific metric spaces.
problem Investigating curvature properties of Cartan spaces with mth root metrics.
method Analyzing isotropic mean Berwald and Landsberg curvatures, and examining conformal β-changes.
result Conditions for conformal β-change of m-th root metrics to be locally dually flat and locally Minkowskian.
The paper classifies Finsler surfaces satisfying the T-condition or σT-condition.
problem Characterizing Finsler surfaces based on specific tensor conditions.
method Analyzing Finsler surfaces in dimensions n≥3, proving conditions equivalence, and solving PDEs.
result All Finsler surfaces satisfying the T-condition or σT-condition are classified.
The paper explores parallel 1-forms on special Finsler manifolds and their properties.
problem Investigating parallel 1-forms on specific Finsler manifolds.
method Analyzing Landsberg manifolds, metrizability freedom, and specific Finsler metrics.
result Landsberg surfaces with parallel 1-forms are necessarily Berwaldian, and the metrizability freedom is at least 2.
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…
The paper proves conjectures about Minkowski norms with specific symmetry groups.
problem Proving conjectures about Minkowski norms with certain symmetries.
method Analyzing isometries of the Hessian metric for Minkowski norms invariant under SO(k)imesSO(n−k). result Proves Laugwitz and Landsberg Unicorn conjectures for Minkowski norms with the specified symmetry.
Study new warped metrics with vanishing Douglas curvature.
problem Warped product metrics and their curvature properties.
method Solve differential equation for vanishing Douglas curvature.
result Classify Douglas warped product 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 -…
Paper proves an equivalence theorem for a class of norms.
problem Investigating equivalence of norms in Finsler geometry.
method Detailed analysis of Cartan tensors and proof of equivalence theorem.
result General (α,β)-metrics on manifolds with vanishing Landsberg curvatures are Berwald manifolds.
Finsler gravity vacuum equation reduces to Ricci vanishing under specific conditions.
problem Solving Einstein vacuum equations in Finsler gravity
method Identifying conditions for vacuum equation reduction
result Scalar Finsler gravity vacuum equation reduces to Ricci vanishing
Paper studies Minkowskian product of Finsler manifolds and their connections.
problem Exploring properties of Minkowskian product of Finsler manifolds.
method Derives Cartan and Berwald connections for the product manifold.
result Necessary and sufficient conditions for the product manifold to have specific properties.
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…
In the previous work, the notion of the Finsleroid--Finsler space have been formulated and the necessary and sufficient conditions for the space to be of the Landsberg type have been found. In the present paper, starting with particular spray coefficients, we demonstrate how the Landsberg condition can explicitly appea…
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 (M,F) is a Finsler structure such that there exists a Riemannian structure that leaves…
Study on special Finsler metrics with conditions for Riemannian and isotropic properties.
problem Characterizing Finsler metrics with specific geometric properties.
method Analyzing conditions for Riemannian and isotropic properties of AR-Finsler metrics.
result Conditions for isotropic S-curvature and mean Landsberg curvature leading to vanishing curvature. New metrics defined in Finsler geometry with specific properties.
problem Understanding the properties of Finsler metrics and their subclasses.
method Introducing the generalized Berwald projective Weyl metric and proving properties of the class of generalized Douglas metrics.
result All GDW metrics with vanishing Landsberg curvature are of R-quadratic type. The paper classifies (α,β)-metrics based on tensor conditions.
problem Classifying (α,β)-metrics based on tensor conditions. method Analyzing (α,β)-metrics for T-tensor vanishing and σT-condition. result Identifies metrics of the first class as Berwaldian and the second class as almost regular non-Berwaldian Landsberg metrics.
This paper addresses the problem of existence of generalized Landsberg structures on surfaces using the Cartan-Kähler Theorem and a Path Geometry approach.