New definition of metric current yields Finsler geometry volume densities.
problem Defining volume functionals from Finsler geometry.
method Proposed a new definition of metric current and showed its utility.
result Obtained a family of extendibly convex volume densities.
In this paper, we mainly establish a Cheeger type finiteness theorem for Berwald manifolds. In order to do this, we study the injectivity radius and the convex radius of a Finsler manifold. A Cheeger type estimate on injectivity radii for Finsler manifolds is given and the existence of the center of mass of a Berwald m…
A method to generalize results from Riemannian Geometry to Finsler geometry is presented. We use the method to generalize several results that involve only metric conditions. Between them we show that the topology induced by the Finsler structure is equivalent to the manifold topology, we provide a new proof of the Hop…
The correspondence between Riemann-Finsler geometries and effective field theories with spin-independent Lorentz violation is explored. We obtain the general quadratic action for effective scalar field theories in any spacetime dimension with Lorentz-violating operators of arbitrary mass dimension. Classical relativist…
The Riemannian hemisphere has a lower bound for its mass.
problem Estimating the mass of surfaces spanning a circle.
method Constructing a differential form with a stationary comass norm on the hemisphere.
result The mass of surfaces spanning a circle has a lower bound of 2π plus a second-order term. We investigate the Cartan and Finsler geometry of the rotating Kepler problem, a limit case of the restricted three body problem that arises if the mass of the one of the primaries goes to zero. We show that the Hamiltonian for the rotating Kepler problem can be regarded as the Legendre transform of a certain family of…
Derives equations of motion for systems with angular momentum on Finsler geometries.
problem Equations of motion for dynamical systems with angular momentum on Finsler geometries.
method Apply Souriau's Principle of General Covariance to derive diffeomorphism invariant equations of motion.
result Generalizes Mathisson-Papapetrou-Dixon equations to Finsler geometries and finds conserved quantities.
Unified framework for photon and massive particle hypersurfaces in stationary spacetimes.
problem Understanding photon and massive particle hypersurfaces in stationary spacetimes.
method Unified framework using Killing-invariant timelike hypersurfaces and associated Finsler structures.
result Conditions for a hypersurface to be a photon or massive particle hypersurface are established.
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.
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. 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.
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if F is a singular Finsler foliation on a Randers manifold (M,Z) with Zermelo data (h,W), then $\mathcal{F}…
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 proves a Schwarz lemma for weakly Kähler-Finsler manifolds.
problem Estimating distance functions and proving Schwarz lemma for weakly Kähler-Finsler manifolds.
method Establishing theorems about distance functions and applying them to prove the Schwarz lemma.
result Holomorphic mappings from weakly Kähler-Finsler manifolds to pseudoconvex Finsler manifolds are constant under certain conditions.
The study introduces new tensors for almost Finsler manifolds and analyzes their properties.
problem Defining and analyzing new types of Finsler manifolds.
method Introducing new Finsler manifolds, studying their properties, and deriving characteristic tensors.
result Characteristic tensors for almost Finsler manifolds have been generalized and their properties have been studied.
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…
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.
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. 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.
The pullback approach to global Finsler geometry is adopted. Some new types of special Finsler spaces are introduced and investigated, namely, Ricci, generalized Ricci, projectively recurrent and m-projectively recurrent Finsler spaces. The properties of these special Finsler spaces are studied and the relations betwee…
The aim of the present paper is to provide an intrinsic investigation of two special Finsler spaces whose defining properties are related to Berwald connection, namely, Finsler space of scalar curvature and of constant curvature. Some characterizations of a Finsler space of scalar curvature are proved. Necessary and su…
The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.
problem Characterizing and understanding harmonic Finsler manifolds.
method Investigation of various types of harmonic Finsler manifolds, characterizations via mean curvature and Laplacian, and construction techniques.
result Certain harmonic Finsler manifolds are of Einstein type and examples of non-Riemannian Finsler harmonic manifolds are provided.
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.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
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.
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…
The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.
problem Constructing Funk-Finsler structures in hyperbolic models.
method Using Finsler isometries and explicit computations, the Funk-Finsler structure is constructed in various hyperbolic models.
result The Funk-Finsler structure in the Klein unit disc is a Randers metric.
The study extends Obata's theorem and classifies Finsler manifolds with transnormal functions.
problem Classifying Finsler manifolds based on geometric properties.
method Extending Obata's theorem and using a second order differential equation.
result Complete Finsler manifolds of positive constant flag curvature are homeomorphic to spheres.
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
The paper shows examples of geodesics switching infinitely often on certain manifolds.
problem Understanding geodesics with infinitely many switches on Finsler and sub-Finsler manifolds.
method Provided examples and explicit structures on Carnot groups, presented a sufficient condition for chattering.
result Geodesics on certain manifolds can exhibit a countable number of switches in arbitrarily small time intervals.
In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…
Study on Kähler Finsler manifolds with curvature bounds, proving theorems.
problem Understanding Kähler Finsler manifolds with curvature constraints.
method Analyzing partial parallelism of complex structure, proving theorems.
result Generalized comparison theorem for positively curved Kähler Finsler manifolds.
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.
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.
We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first eigenvalue of the Finsler-Laplacian attains its lower bound if and only if the Fi…
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.
The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.
problem Existence and uniqueness of Ricci flow solutions on Finsler manifolds.
method Investigation of short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds.
result Theorems demonstrating the short-time existence of the flow solution for n-dimensional Finsler manifolds and the uniqueness of the solution for isotropic Finsler manifolds.
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.
Extends Penrose limit to Finsler spacetimes.
problem Extending Penrose limit to Finsler spacetimes.
method Introducing lightlike coordinates and adapting Lorentzian pp-wave definition.
result New examples of Finsler pp-waves presented.
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. 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 study examines Einstein-Finsler spaces using Minkowskian products.
problem Characterizing Einstein-Finsler spaces through Minkowskian products.
method Proving conditions for Einstein-Finsler spaces in terms of Minkowskian products.
result If a Minkowskian product Finsler manifold is Einstein, then either the product manifold is Ricci flat or both quotient manifolds are Einstein with same scalar functions.
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 …
The paper studies new curvature properties in Finsler geometry.
problem Properties of projectively equivalent Finsler metrics and their curvature structures.
method Introducing new characterizations of quadratic curvature properties in Finsler manifolds.
result Novel insights into curvature behavior under generalized projective sprays.
Two special Finsler spaces have been introduced and investigated, namely Rh-recurrent Finsler space and consircularly recurrent Finsler space. The defining properties of these spaces are formulated in terms of the first curvature tensor of Cartan connection. The following three results constitute the main object of …
Generalizes O'Neill's equations to pseudo-Finsler submersions.
problem Extending Riemannian submersion equations to pseudo-Finsler geometry.
method Generalization of O'Neill's fundamental equations to pseudo-Finsler submersions and exploration of O'Neill tensors.
result Generalized fundamental equations for pseudo-Finsler submersions.
Here, a non-linear analysis method is applied rather than classical one to study projective Finsler geometry. More intuitively, by means of an inequality on Ricci-Finsler curvature, a projectively invariant pseudo-distance is introduced and an analogous of Schwarz' lemma in Finsler geometry is proved. Next, the Schwarz…