Paper examines conditions for singular square metrics to have constant curvature.
problem Conditions for constant curvature in singular square metrics.
method Analyzes Finsler metrics, introduces singular square metrics, provides necessary and sufficient conditions.
result Necessary and sufficient conditions for constant Ricci or flag curvature in singular square metrics.
Characterizes Finsleroid-Finsler metrics as solutions to a conformal rigidity problem.
problem Existence of regular Landsberg metrics that are not of Berwald type.
method Characterizes Finsleroid-Finsler metrics as solutions to a conformal rigidity problem involving a specific invariant curvature tensor.
result Solutions of class at least C2 on the complement of the zero section are conformal to Finsleroid-Finsler metrics. Investigates singular Finsler foliations on (α,β)-spaces and their relation to Riemannian foliations.
problem Understanding conditions for singular Finsler foliations to be singular Riemannian foliations.
method Analyzes (α,β)-spaces and verifies conditions for SFFs to be SRFs, extending Molino's conjecture. result Equifocality of regular leaves for SFFs under certain conditions.
Singular Finsler metrics, such as Kropina metrics and m-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric α and 1-form β and characterize those which are respectively Douglasian and locally projectively flat in di…
The paper proves properties of Finsler submanifolds and analytic maps.
problem Analyzing properties of Finsler submanifolds and their analytic maps.
method Proving properties of regular fibers of analytic maps and Finsler submersions.
result Regular fibers of an analytic map are equifocal under certain conditions.
Singular Finsler metrics, such as Kropina metrics and m-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular (α,β)-metrics which are locally projectively flat with constant flag curvature in dimension n=2 and n≥3 respectively. Further, we determine t…
Singular Finsler metrics, such as Kropina metrics and m-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of two-dimensional singular Finsler metrics defined by a Riemann metric α and 1-form β, and we characterize those which are Douglasian or locally projectively flat…
Paper characterizes square metrics with vanishing Douglas curvature.
problem Characterizing square metrics with vanishing Douglas curvature.
method Using β-deformations to achieve analytical examples.
result Characterization of singular square metrics with vanishing Douglas curvature.
We give a Finsler non-linear connection by a new simplified definition for not only regular case but also singular case. In regular case, it corresponds to non-linear connection part of Berwald's connection, but our connection is expressed not in line element space but in point-Finsler space. In this view we recognize …
Paper classifies positively curved δ-homogeneous Finsler metrics.
problem Classifying positively curved Finsler metrics.
method Exploring δ-homogeneity and using approximation techniques.
result Classification of positively curved δ-homogeneous Finsler spaces.
Introduces singular Finsler foliations and their properties.
problem Generalizing Finsler actions, submersions, and foliations.
method Definition and analysis of singular Finsler foliations on Randers manifolds.
result Singular Finsler foliations on Randers manifolds are singular Riemannian foliations on the Riemannian manifold induced by the metric.
In this paper we study the flag curvature of a particular class of Finsler metrics called general (α,β)-metrics, which are defined by a Riemannian metric α and a 1-form β. The classification of such metrics with constant flag curvature are completely determined under some suitable conditions, which make them be…
Paper proves Asanov's Unicorns are generalized Berwald manifolds.
problem Existence of regular Landsberg metrics that are not of Berwald type.
method Proves Finsleroid-Finsler spaces are generalized Berwald spaces under certain conditions.
result Asanov's Unicorns belong to the class of generalized Berwald manifolds.
In this paper, we consider a special class of singular Finsler metrics: m-Kropina metrics which are defined by a Riemannian metric and a 1-form. We show that an m-Kropina metric (m=−1) of scalar flag curvature must be locally Minkowskian in dimension n≥3. We characterize by some PDEs a Kropina metric ($…
Study on conditions for Randers metrics to be of constant Ricci curvature.
problem Conditions for Randers metrics to be of constant Ricci curvature.
method Analysis of sufficient and necessary conditions for Randers metrics with and without strong convexity.
result Classification of Randers metrics with ∥β∥α>1 and ∥β∥α≡1. The abstract develops weighted Ricci curvature in Lorentz-Finsler geometry and extends singularity theorems.
problem Extending singularity theorems in weighted Lorentz-Finsler geometry.
method Generalizing Jacobi, Riccati, and Raychaudhuri equations; applying generalized Bishop inequality.
result Weighted Lorentz-Finsler singularity theorems extended.
Using the definition of entropy of a family of increasing distances on a compact metric set given in [10] we introduce a notion of Finsler entropy for smooth distributions and Stefan-Sussmann foliations. This concept generalizes most of classical topological entropy on a compact Riemannian manifold : the entropy of a f…
We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized…
Describes links between Finsler and Lorentz geometries for Riemannian geometers.
problem Understanding the relationship between Finsler and Lorentz geometries.
method Analyzes the Zermelo navigation problem and develops issues related to causality, Finsler elements, and wave propagation.
result Provides a comprehensive understanding of the Lorentzian causality using Finsler elements and the natural relation between the Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler setting.
The paper defines a metric on Euclidean triangles and polygons, proving properties and completeness.
problem Defining and analyzing a metric space for Euclidean triangles and polygons.
method Introducing and proving properties of a metric on marked Euclidean triangles, extending to polygons and triangulated surfaces.
result The metric is Finsler and complete, providing formulas for its infinitesimal structure.
Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the origin…
Characterizes complex Finsler metrics and their properties.
problem Characterize complex Finsler metrics and their geometric properties.
method Defined the canonical connection and investigated holomorphic sectional curvature tensors and Ricci curvatures.
result Characterizes balanced complex Finsler metrics and provides sufficient and necessary conditions.
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. 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.
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.
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 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. New Finsler metrics constructed from GDW-metrics.
problem Exploring new Finsler metrics within the GDW-metric class. method Constructing new sub-classes of GDW-metrics. result Presented illustrative examples of new Finsler metrics.
Classifies reversible Finsler metrics with positive curvature.
problem Classifying homogeneous reversible Finsler metrics with positive flag curvature.
method Classification based on G-invariant metrics and curvature properties.
result Exceptions exist where homogeneous Finsler metrics with positive flag curvature are not known.
Investigate AR-Finsler metrics for local dual flatness and projective flatness.
problem Locally dually and projectively flat AR-Finsler metrics
method Derive necessary and sufficient conditions and a compatibility relation.
result Establish a rigidity result for AR-Finsler metrics.
Study Ricci curvature of homogeneous Finsler spaces with specific metrics.
problem Curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Derived explicit formulae for Ricci curvature and found conditions for vanishing S-curvature. result Spaces with vanishing S-curvature and negative Ricci curvature are Riemannian. New definition of naturally reductive Finsler manifolds using geodesic graphs.
problem Defining naturally reductive Finsler manifolds using geodesic graphs.
method Proposed a new geometrical definition using geodesic graphs and constructed examples of Finsler metrics.
result Explicit examples of Finsler naturally reductive metrics constructed.
A theorem on Finsler metrics with special curvature properties.
problem Characterizing Finsler metrics with sectional flag curvature.
method Analyzing the relationship between flag curvature and sectional curvature.
result Locally, Finsler metrics are either Riemannian or have isotropic flag curvature.
New approach connects Finsler geometry's metric and connections.
problem Deriving Finsler geometry's metric and connections from compatibility axioms.
method Compatibility axioms between metric and Finsler connection.
result Metrical formulation of Finsler geometry for field theory.
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.
Study cylindrical symmetric Finsler metrics with vanishing Douglas curvature.
problem Understanding Finsler metrics with specific curvature properties.
method Analyzing cylindrically symmetric Finsler metrics and solving differential equations.
result Differential equations for cylindrically symmetric Finsler metrics with vanishing Douglas curvature.
This paper explores new Finsler metrics and their connections to generalized Sakaguchi's Theorem.
problem Understanding the properties of Finsler metrics and their projective invariants.
method Developed and examined weakly-Weyl and generalized weakly-Weyl Finsler metrics.
result Equivalence of weakly-Weyl and W-quadratic spherically symmetric Finsler metrics. We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we …
In 2D, Finsler metrics with 3+ projective fields are projectively equivalent to Randers.
problem Characterizing Finsler metrics with multiple projective vector fields.
method Analyzing the Lie algebra of projective vector fields and showing equivalence to Randers metrics.
result A complete list of 2D Finsler metrics with at least 3 projective fields up to equivalence.
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. Study flag curvature in homogeneous Finsler spaces with a specific metric.
problem Analyzing flag curvature in homogeneous Finsler spaces with a generalized m-Kropina metric. method Provided explicit formula for flag curvature, showed equivalence of definitions, and studied curvature of naturally reductive spaces.
result Equivalence of two definitions of naturally reductive homogeneous Finsler spaces for the generalized m-Kropina metric. The paper studies special metrics on homogeneous Finsler spaces and calculates curvature.
problem Calculating curvature for specific metrics on homogeneous Finsler spaces.
method Proved existence of invariant vector fields and derived formulas for S-curvature and mean Berwald curvature.
result Explicit formulas for S-curvature and mean Berwald curvature derived for specific metrics.
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
The paper examines Finsler metrics under Ricci flow and finds they are Einstein.
problem Understanding Finsler metrics under Ricci flow.
method Investigated C3-like Finsler metrics and proved they are Einstein.
result C3-like Finsler metrics are Einstein under Ricci flow.
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.
Study on conformal deformations of complex Finsler metrics.
problem Characterization and stability of Kähler Finsler metrics.
method Characterization and study of critical points in conformal classes.
result Stability of critical Kähler Finsler metrics obtained.
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…