The paper studies Finsler spaces with semi-concurrent vector fields and their equivalence to Riemannian spaces.
problem Characterizing Finsler spaces with semi-concurrent vector fields.
method Analyzing various Finsler spaces and proving conditions for equivalence to Riemannian spaces.
result Various Finsler spaces (quasi-C-reducible, C3-like, Ch-recurrent, P2-like) are equivalent to Riemannian spaces if they admit a semi-concurrent vector field. The paper defines semi-concurrent vector fields on Finsler manifolds and explores their implications.
problem Characterizing Finsler manifolds with semi-concurrent vector fields.
method Introducing and investigating semi-concurrent vector fields, proving properties, and providing examples.
result Finsler manifolds admitting semi-concurrent vector fields are either Riemannian or conic Finslerian.
Study proves conformal vector fields on certain Finsler manifolds are Killing fields.
problem Characterizing conformal vector fields on compact homogeneous Finsler manifolds.
method Analyzes properties of conformal vector fields and homogeneous Finsler metrics.
result Conformal vector fields on compact homogeneous Finsler manifolds are Killing fields.
Study connects Riemann-Finsler geometry to Lorentz-violating scalar fields.
problem Exploring the connection between Riemann-Finsler geometries and Lorentz-violating scalar fields.
method Deriving quadratic actions and classical relativistic point-particle lagrangians in various spacetime dimensions.
result Support for open conjectures about Riemann-Finsler geometries in Lorentz-violating field theories.
Zermelo deformation preserves geodesics and curvature in Finsler metrics.
problem Behavior of geodesics and curvature in Finsler metrics under Zermelo deformation.
method Zermelo deformation with Killing vector fields.
result Zermelo deformation preserves local symmetry in locally symmetric Finsler metrics.
Mathematical framework for field theories on Finsler spacetimes.
problem Developing a consistent calculus for field theories on Finsler spacetimes.
method Constructing configuration bundles and applying coordinate-free calculus of variations.
result Averaged energy-momentum conservation law for Finsler field theories.
Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.
problem Exploring projective symmetries in Finsler spaces and their properties.
method Analyzing algebraic sub-algebras and curvature invariants of projective vector fields.
result Closed Finsler spaces with negative Ricci curvature reduce to Killing vector fields.
Study on Finsler spacetimes with timelike Killing vectors, introducing stationary splitting.
problem Characterizing Finsler spacetimes with timelike Killing vectors.
method Introducing a new class of stationary splitting Finsler spacetimes and characterizing their properties.
result A Finsler spacetime with a timelike Killing vector field is locally a stationary splitting.
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 Finsler metrics with Killing fields on constant flag curvature surfaces.
problem Characterize Finsler metrics with Killing fields on surfaces of constant flag curvature.
method Developed a normal form and method to calculate functions for spherically symmetric Finsler surfaces.
result Obtained the normal form of the Funk metric on the unit disk D^2.
Study of conformal mappings in pseudo-Finsler spaces, extending results from pseudo-Riemannian geometry.
problem Understanding conformal mappings in pseudo-Finsler spaces and their properties.
method Extending results from pseudo-Riemannian geometry and proposing a technique to reduce problems involving pseudo-Finslerian conformal vector fields.
result Conformal groups of flat pseudo-Finsler spaces can be richer than flat Finslerian and pseudo-Euclidean conformal groups.
The aim of the present paper is to investigate intrinsically the notion of a concircular π-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a concurrent vector field in Finsler geometry. Some properties of concircular π-vector fie…
New definitions and properties of harmonic vector fields on Finsler manifolds.
problem Defining and understanding harmonic vector fields in Finsler geometry.
method Natural definitions of differential, divergence, and p-harmonic form; proving Hodge theorem; Bochner-Yano classification theorem. result A closed orientable Finsler manifold with a positive harmonic Ricci scalar has a zero Betti number.
Simplified proof for affine vector fields on Finsler manifolds.
problem Generalization of Hano's theorem to Finsler manifolds.
method Simpler proof for a more general case.
result Affine vector fields on complete Finsler manifolds are Killing fields.
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.
Characterizes affine vector fields on Finsler manifolds with rigidity results.
problem Understanding affine vector fields on Finsler manifolds.
method Utilizing the Jacobi type equation and spray characterization, proving rigidity theorems.
result Rigidity theorems for affine vector fields on Finsler manifolds with non-positive total Ricci curvature.
The paper proves a monodromy theorem for sheaves of local fields, extending previous results.
problem Proving a monodromy theorem for sheaves of local fields.
method Proves a monodromy theorem for local vector fields in sheaves satisfying the unique continuation property.
result Obtains a global extension result for every local field of the sheaf.
Introduces new geodesic fields for Finsler manifolds.
problem Finding optimal paths in Finsler manifolds.
method Introduced Pontryagin type C0-Finsler structures and defined geodesic fields. result Extended geodesic field E provides more natural geodesics. In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metri…
We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.
The paper discusses fundamental problems in global Finsler geometry.
problem Fundamental problems in global Finsler geometry.
method Optimization and improvement of Lie derivatives, characterization of gradient vector fields, gradient estimate.
result Gradient estimate for smooth functions on Randers manifolds.
Study examines causal properties of Finsler spacetimes with cone Killing vectors.
problem Characterize causality in Finsler spacetimes with specific Killing vectors.
method Explores the relationship between wind Riemannian structures and spacetimes with cone Killing vectors, focusing on Finsler-Kropina metrics.
result Characterizes causality properties using metric-type properties of Finslerian structures.
Defines a new type of Finsler space with specific metrics.
problem Defines a new type of Finsler space with specific metrics.
method Defines and studies a new type of Finsler space with Infinite Series (α,β)-metric. result Properties of the new Finsler space are studied.
Paper studies conformal vector fields in specific Finsler spaces.
problem Characterizing and understanding conformal vector fields in (α,β) spaces. method Using principles to characterize and determine local structure of conformal vector fields under curvature conditions.
result Construction of non-homothetic conformal vector fields on locally projectively Randers spaces.
It is proved that all invariant functions of a complex Finsler manifold can be totally recovered from the torsion and curvature of the connection introduced by Kobayashi for holomorphic vector bundles with complex Finsler structures. Equations of the geodesics and Jacobi fields of a generic complex Finsler manifold, ex…
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.
Applying concepts and tools from classical tangent bundle geometry and using the apparatus of the calculus along the tangent bundle projection ('pull-back formalism'), first we enrich the known lists of the characterizations of affine vector fields on a spray manifold and conformal vector fields on a Finsler manifold. …
The paper studies geodesic circles and concircular transformations in Finsler geometry.
problem Characterizing and understanding concircular transformations in Finsler geometry.
method Applying Lie derivatives and the Cartan Y-connection to study geodesic circles and concircular transformations. result Characterization of concircular vector fields and conditions for conformal and concircular transformations.
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.
Finsler geometry connects to higher-spin fields and curved-space Fronsdal equations.
problem Eliminating non-transverse modes in Finsler dynamics for higher spins.
method Parameterizing Finsler geometry in terms of symmetric tensors, analyzing linear and nonlinear terms, and examining gauge transformations.
result Finsler dynamics leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling.
Study of Randers spacetimes and their Finsler gravity solutions.
problem Analyzing Finsler gravity field equations for Randers spacetimes.
method Examined Berwald-type Randers spacetimes and Finsler gravity field equations, showing equivalence to Einstein gravity.
result Found exact solutions for vacuum Finsler gravity are composed of pp-waves and 1-forms.
The paper introduces new metrics on Finsler manifolds and characterizes associated vector fields.
problem Characterizing vector fields on Finsler manifolds with new metrics.
method Introducing F-natural metrics and characterizing conformal, homothetic, and Killing vector fields. result Characterization of vector fields on slit tangent bundles of Finsler manifolds.
The paper explores traveling along broken geodesics in Finsler submersions.
problem Analyzing the attainable sets of analytic vector fields in Finsler submersions.
method Investigates the dual leaves and attainable sets of horizontal broken geodesics.
result Proves that in compact Finsler manifolds with positive flag curvature, the attainable sets coincide with orbits.
The paper constructs Einstein Finsler metrics on Riemannian manifolds.
problem Characterizing Einstein Finsler metrics on Riemannian manifolds.
method Using a Killing form, the paper constructs a class of Finsler metrics and finds equations for Einstein metrics.
result The paper provides a family of Einstein metrics on S3 with specific Ricci curvature properties. Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …
The π-exterior derivative ød, which is the Finslerian generalization of the (usual) exterior derivative d of Riemannian geometry, is defined. The notion of a ød-closed vector field is introduced and investigated. Various characterizations of ød-closed vector fields are established. Some results concerning $ød…
The present paper deals with the Killing correspondence between some Finsler spaces. We consider a Finsler space equipped with a β-change of metric and study the Killing correspondence between the original Finsler space and the Finsler space equipped with β-change of metric. We obtain necessary and sufficient condi…
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
This study compares Riemannian and Finsler geometries, focusing on their cut and conjugate loci.
problem Comparing Riemannian and Finsler geometries, focusing on their cut and conjugate loci.
method Discussion of fundamental results on cut and conjugate loci of Finsler manifolds, highlighting differences from Riemannian manifolds.
result Differences in cut and conjugate loci between Riemannian and Finsler manifolds.
Survey on positively curved homogeneous Finsler spaces.
problem Classifying positively curved homogeneous Finsler spaces.
method Classification and related topics.
result Presentation of open problems in the field.
Analyzing static solutions in Finsler gravity, extending known results.
problem Extending the analyticity of static vacuum solutions to Finsler spacetimes.
method Examining Finsler spacetimes with properties similar to static Lorentzian spacetimes.
result Finsler spacetimes with vanishing Ricci scalar are analytic.
We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric F is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of F by a vector field W at each point, where W is an arbitrary …
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.
We generalize Matsumoto metrics with a special π-form and explore their geometric properties.
problem Exploring the geometric properties of generalized Matsumoto metrics with a special π-form.
method Considering a Finsler manifold with a concurrent π-vector field, we introduce a change in the metric and analyze its geometric properties.
result The generalized φ-Matsumoto metric can never be projectively related to the original metric.
New exact spherically symmetric vacuum solutions found in Finsler gravity.
problem Finding exact vacuum solutions in Finsler gravity.
method Spherically symmetric, asymptotically flat Berwald spacetimes solved for Finsler gravity vacuum equation.
result Only one class of spherically symmetric Berwald spacetimes is compatible with asymptotic flatness and a well-defined causal structure.
Paper derives a formula for Finsler manifolds using superconnections.
problem Developing a formula for Finsler manifolds without vector fields.
method Using Mathai-Quillen's superconnection formalism and geometric localization.
result Established an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds.
Paper proves geodesics and focal points unchanged by conformal changes in pseudo-Finsler manifolds.
problem Understanding the behavior of lightlike geodesics in pseudo-Finsler manifolds.
method Used Chern connection, anisotropic calculus, and critical points of energy functional to prove invariance.
result Lightlike geodesics and focal points are preserved by anisotropic conformal changes.
In this paper, we study Clifford-Wolf translations of Finsler spaces. We first give a characterization of Clifford-Wolf translations of Finsler spaces in terms of Killing vector fields. In particular, we show that there is a natural correspondence between Clifford-Wolf translations and the Killing vector fields of cons…