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.
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…
Study classifies Finsler metrics with isotropic Berwald curvature.
problem Classifying Finsler metrics with isotropic Berwald curvature.
method Defined by a Riemannian metric and a 1-form, classified under specific conditions.
result Classified general (α,β)-metrics with isotropic Berwald 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.
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.
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.
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 …
Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.
problem Investigating curvature properties in Finsler geometry.
method Examined the geometry of fibres and proved curvature constancy under isotropy.
result Berwald scalar curvature is constant along fibres when mean Berwald curvature is isotropic.
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. 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 study the second approximate Matsumoto metric on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.
New sprays of constant curvature introduced; conditions for metrizability given.
problem Conditions for sprays of isotropic curvature to be metrizable.
method Introducing sprays of constant curvature and investigating metrizability conditions.
result Sprays of isotropic curvature are not necessarily of constant curvature.
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 explores generalized Douglas-Weyl metrics and their properties.
problem Characterizing and understanding generalized Douglas-Weyl metrics.
method Analyzing properties of (α,β)-metrics and proving conditions for being generalized Douglas-Weyl metrics. result Generalized Douglas-Weyl metrics are Berwald metrics under certain conditions.
Study on spherical Finsler metrics with isotropic curvature rigidity.
problem Characterizing and understanding spherically symmetric Finsler metrics with isotropic E-curvature. method Provided the correct formula for mean Berwald curvature, established differential equations for projective and dual flatness, and derived a rigidity result.
result Rigidity result on spherically symmetric Finsler metrics with isotropic E-curvature. In this paper, we investigate the spherically symmetric Finsler metrics with isotropic S-curvature and obtain a characterized equation. As an application, we prove that these metrics with Douglas type must be Randers metrics or Berwald metrics. This result leads to two classification theorems.
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 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…
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…
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.
Researchers generalize cosmological models using Finsler geometry.
problem Cosmological models that closely approximate pseudo-Riemannian geometry.
method Identifying Lie Algebra of symmetry generators for spatially homogeneous and isotropic Finsler geometries.
result Found the most general spatially homogeneous and isotropic Berwald spacetimes.
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.
Study Berwald spacetimes and their relation to very special relativity.
problem Understanding vacuum dynamics in Berwald spacetimes.
method Derived Finsler generalization of Einstein's equations and derived conditions for Berwald spacetimes.
result Identified necessary and sufficient conditions for VGR line elements to be Berwald type.
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 le…
Berwald spaces with non-zero flag curvature are Riemannian.
problem Understanding the conditions under which Berwald spaces become Riemannian.
method Analyzing the flag curvature of Berwald spaces and proving their rigidity under certain conditions.
result Berwald spaces with non-zero flag curvature are Riemannian.
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.
Higher rank rigidity theorem extended to Berwald spaces.
problem Generalizing rigidity theorem to Berwald spaces.
method Proving rigidity for Berwald spaces with specific curvature conditions.
result Berwald spaces with certain conditions are locally symmetric or Minkowski.
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.
Study on generalized m-Kropina metrics in modified gravity and cosmology.
problem Understanding the geometric properties and applications of generalized m-Kropina metrics. method Proving the rationality of Finslerian geometric objects and studying the conditions for Einstein metrics.
result Conditions for a generalized m-Kropina metric to be an exact solution in modified gravity and cosmology. In this note it is shown that Berwald spaces admitting the same norm-preserving torsion-free affine connection have the same (weighted) Ricci curvatures. Combing this with Szabó's Berwald metrization theorem one can apply the Cheeger-Gromoll splitting theorem in order to get a full structure theorem for Berwald spaces …
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.
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…
In this paper by using left invariant Riemannian metrics on some 3-dimensional Lie groups we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally Minkowskian spaces of zero constant flag curvature.
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.
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. 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.
Study pseudo-Finsler spaces on pseudo-Minkowski space, deriving geometric objects.
problem Characterize geometric properties of pseudo-Finsler spaces.
method vierbein formalism, derive closed expressions for geometric objects.
result Non-trivial Berwald spaces have indicatrices with non-trivial symmetries.
Classifies surfaces in isotropic geometry with constant curvature.
problem Classifying surfaces with constant curvature in isotropic geometry.
method Classifies surfaces under the condition that at least one translating curve lies in a plane.
result Classification of surfaces in isotropic geometry with constant curvature.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic S-curvature and are either Minkowskian or Riemannian. 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.
Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…
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.
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.
Study finds first integrals in Finsler metrics with vanishing χ-curvature.
problem Analyzing Finsler metrics with vanishing χ-curvature.
method Proving geodesic invariance of certain structures and deriving first integrals.
result Induces a set of first integrals in Finsler metrics with vanishing χ-curvature.
The paper studies anisotropic conformal changes in pseudo-Finsler surfaces.
problem Investigating geometric properties under anisotropic conformal transformations.
method Analyzing geometric objects like Berwald, Landsberg, and Douglas tensors under anisotropic conformal transformations.
result Necessary and sufficient conditions for various geometric properties under anisotropic conformal transformations.
The paper characterizes spherically symmetric metrics with scalar curvature.
problem Characterizing spherically symmetric metrics with scalar curvature.
method Established a curvature compatibility condition on spherically symmetric Finsler metrics and constructed a Berwald frame.
result Characterized spherically symmetric metrics with scalar curvature.
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. In this paper we introduce a natural definition for the affine maps between two Finsler manifolds (M,F) and (N,F~) and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;…