In 2D, Finsler metrics are Douglas and generalized Berwald if they are Berwald or Randers.
problem Characterizing Finsler metrics in 2D that are both Douglas and generalized Berwald.
method Proof that in dimension two, a Finsler metric is both Douglas and generalized Berwald if and only if it is Berwald or a Randers metric α+β with specific properties. result Finsler metrics in 2D are Douglas and generalized Berwald if they are Berwald or Randers metrics with specific properties.
The abstract proves properties of Berwald spaces with non-zero flag curvature.
problem Characterizing Berwald spaces with non-zero flag curvature.
method Analyzes properties of Berwald manifolds with non-zero flag curvature, proving extensions of previous theorems.
result Every Berwald manifold with non-zero flag curvature is Riemannian.
Characterizes two-dimensional generalized Berwald metrics with vanishing S-curvature.
problem Characterizing metrics with specific curvature properties.
method Analyzing two-dimensional generalized Berwald (α,β)-metrics with vanishing S-curvature. result Provides a generalization of Szabó rigidity theorem for (α,β)-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; …
Characterizes and simplifies m-th root Finsler metrics.
problem Understanding and simplifying m-th root Finsler metrics. method Characterization and reduction of m-th root Finsler metrics. result Every isotropic mean Berwald curvature m-th root Finsler metric reduces to a weakly Berwald metric. Monochromatic Finsler metrics are shown to be generalized Berwald metrics.
problem Characterizing Finsler metrics with isomorphic tangent spaces.
method Demonstrating the existence of an affine connection preserving the Finsler function.
result Monochromatic Finsler metrics are generalized Berwald metrics.
Characterizes Kähler-Berwald metrics on complex manifolds.
problem Identifying Kähler-Berwald metrics among strongly convex complex Finsler metrics.
method Geometric characterization using Cartan and Chern-Finsler connections.
result Characterizes Kähler-Berwald metrics in terms of parallelism of the canonical complex structure.
Berwald metrics are particular Finsler metrics which still have linear Berwald connections. Their complete classification is established in an earlier work, [Sz1], of this author. The main tools in these classification are the Simons-Berger holonomy theorem and the Weyl-group theory. It turnes out that any Berwald metr…
Study on Berwald-Weyl curvature with projective invariance and vanishing results.
problem Characterizing Berwald-Weyl curvature for spray/Finsler metrics.
method Analyzing expressions and proving vanishing conditions for curvature.
result Berwald-Weyl curvature vanishes for certain spray/Finsler metrics.
In this paper, we study generalized Douglas-Weyl (α,β)-metrics. Suppose that an regular (α,β)-metric F is not of Randers type. We prove that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover by ignoring the regularity, if F is not a Berwald met…
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
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.
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 man…
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 metr…
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. 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.
The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.
problem Rigidity of non-positively curved homogeneous Finsler metrics.
method Analyzes and proves rigidity results for specific Finsler metrics with non-positive flag curvature.
result Homogeneous Finsler spaces with non-positive flag curvature and isotropic S-curvature are either Riemannian or locally Minkowskian.
The aim of this paper is to develop on the 1-jet space J^1(R,M^4) the Finsler-like geometry (in the sense of d-connection, d-torsions and d-curvatures) of the rheonomic Berwald-Moor metric. A natural geometrical gravitational field theory produced by the rheonomic Berwald-Moor metric is also constructed.
Paper generalizes Schwarz lemma for polydisc mappings with specific metrics.
problem Schwarz lemma for holomorphic mappings with invariant metrics.
method Analyzes mappings between polydiscs with $\mbox{Aut}(P_m)$-invariant Kähler-Berwald metrics.
result Generalizes Schwarz lemma for polydisc mappings with specific 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 -…
Study on flag curvatures of specific metrics on Lie groups.
problem Investigate flag curvatures of left invariant (α,β)-metrics and Randers metrics. method Explicit formulas and necessary/sufficient conditions for Berwald and Douglas types.
result Flag curvatures of (α,β)-metrics and Randers metrics on Lie groups can be zero, positive, or negative. In the present paper we study Randers metics of Berwald type on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulas for computing flag curvature of th…
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].
In this paper, we study a class of Finsler metrics called general (α,β)-metrics, which are defined by a Riemannian metric α and a 1-form β. We classify this class of Finsler metrics with isotropic Berwald curvature under certain condition.
Defines a new Randers metric based on an existing one.
problem No specific problem stated; focuses on defining a new metric.
method Defines a new left-invariant Randers metric ildeF based on an existing one F. result Shows that F is of Berwald (Douglas) type if and only if ildeF is of Berwald (Douglas) type. The study characterizes Finsler metrics and proves their rigidity.
problem Characterizing and proving rigidity of Busemann convex Finsler metrics.
method Proving Finsler metrics are nonpositively curved if and only if they are affinely equivalent to a Riemannian metric of nonpositive sectional curvature.
result Finsler metrics are precisely Berwald metrics of nonpositive flag curvature.
In this paper, we find a condition on (α,β)-metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.
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 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.
Jebsen-Birkhoff theorem extended to Berwald spacetimes.
problem Generalizing Birkhoff theorem to Berwald spacetimes.
method Proving Ricci-flat, spatially spherically symmetric Berwald spacetimes are pseudo-Riemannian or flat.
result Jebsen-Birkhoff theorem extended to Berwald spacetimes.
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.
In this paper we parametrize the symmetry group of the n-dimensional Berwald-Moor metric. Some properties of this Lie group are studied, and its corresponding Lie algebra is computed.
In this paper we construct the differential equations of the stream lines that characterize plasma regarded as a non-isotropic medium geometrized by a jet rheonomic time-invariant Berwald-Moor metric. Section 1 contains historical notes regarding the Plasma Physics and its geometrical description. Section 2 analyzes th…
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. In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas explicitly.
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…
New findings show Berwald Finsler spacetimes cannot be metrized.
problem Can Szabo's metrizability theorem be extended to Finsler spacetimes?
method Investigation of Berwald Finsler spacetimes and their properties.
result Found a class of Berwald spacetimes with asymmetric Ricci tensors.
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.
Harmonic coordinates for Finsler manifolds prove a theorem but not optimal regularity.
problem Proving the Myers--Steenrod theorem for Finsler manifolds.
method Existence of harmonic coordinates for nonlinear Finsler Laplacian.
result Partial results on optimal regularity for Berwald metrics.
The paper derives formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
problem Understanding linear connections with specific torsion in generalized Berwald manifolds.
method Averaging of Finslerian quantities over indicatrix surfaces to express the torsion tensor.
result Explicit formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
Calculates S-curvature and mean Berwald curvature in homogeneous Finsler spaces.
problem Calculating curvatures in homogeneous Finsler spaces.
method Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.
result Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.
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.
PhD dissertation on Finsler geometry and gravity, focusing on Berwald spaces and exact solutions.
problem Characterizing and solving Finsler gravity equations.
method Analysis of Berwald spaces, (α,β)-metrics, and exact solutions to Finsler gravity equations. result Exact vacuum solutions in Finsler gravity.
We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is clo…
Study on Randers metrics on tangent Lie groups and their geometric properties.
problem Characterizing Randers metrics of Berwald type on tangent Lie groups.
method Analyzing the relations between flag curvature and sectional curvature.
result Identifying all 3-dimensional Lie groups with tangent bundles admitting Berwald type Randers metrics.
Study curvature of complex Finsler metrics on Lie groups.
problem Explicitly calculate and characterize curvature of complex Finsler metrics.
method Analyzes left-invariant complex Finsler metrics on Lie groups using their Lie algebra.
result Characterizes conditions for a Finsler metric to be Kähler or weakly Kähler.
The paper classifies specific types of metrics on 5D Lie groups.
problem Classifying (α,β)-metrics on five dimensional nilpotent Lie groups. method Left-invariant metrics and vector fields classification.
result Geometric properties of classified metrics.
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.