arXiv research

The paper proves inequalities for optimal transport on sub-Finslerian manifolds.

Classification of Finslerian spaces with nontrivial concircular transformations.

Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.

Compact Finslerian manifolds don't admit non-trivial circle-preserving transformations.

In this work, convergence of evolving Finslerian metrics first in a general flow next under Finslerian Ricci flow is studied. More intuitively it is proved that a family of Finslerian metrics $g(t)$ which are solutions to the Finslerian Ricci flow converge in $C^{\infty}$ to a smooth limit Finslerian metric as $t$ ap…

This paper considers fundamental issues related to Finslerian isometries, submetries, distance and geodesics. It is shown that at each point of a Finsler manifold there is a distance coordinate system. Using distance coordinates, a simple proof is given for the Finslerian version of the Myers-Steenrod theorem and for t…

The Hopf-Rinow theorem is extended to sub-Finslerian geometry.

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinat…

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class $Ω$…

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

The paper examines parallel one forms on Riemannian and Finslerian manifolds.

Introduces Finslerian convolution metrics and their properties.

Lecture notes on Finslerian geometry.

Finslerian graph neural networks recover nonlinear diffusion geometry

Paper develops methods for estimating gradients of Finslerian Schrödinger equations.

New findings on lightconvex boundaries in Finslerian spacetimes.

We have obtained Finslerian Ressiner-Nordstrom solution where it is asymptotic to a Finsler spacetime with constant flag curvature while $r\rightarrow\infty$. The covariant derivative of modified Einstein tensor in Finslerian gravitational field equation for this solution is conserved. The symmetry of the special Finsl…

The purpose of this article is to provide a general overview of curvature functional in Finsler geometry and use its information to introduce the gradient flow on Finsler manifolds. For this purpose, we first prove that the space of Finslerian metrics is a Riemannian manifold. Then it is given a decomposition for the t…

Study on nonholonomic mechanics and sub-Finsler geometry.

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Rie…

Let $R$ be the $hh$-curvature associated with the Chern connection or the Cartan connection. Adopting the pulled-back tangent bundle approach to the Finslerian Geometry, an intrinsic characterization of $R$-Einstein metrics is given. Finslerian metrics which are locally conformally $R$-Einstein are classified.

Researchers prove constant solutions for a specific Finslerian equation.

Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact …

We present a new proof of a Finslerian version of Beltrami's theorem (1865) which works also in dimension 2.

Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian…

Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.

Here, an axiom of spheres in Finsler geometry is proposed and it is proved that if a Finslerian manifold satisfies the axiom of spheres then it is of constant flag curvature.

Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.

The notion of wind Finslerian structure is developed; this is a generalization of Finsler metrics where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which …

We introduce the notion of a standard static Finsler spacetime where the base is a Finsler manifold. We prove some results which connect causality with the Finslerian geometry of the base extending analogous ones for static and stationary Lorentzian spacetimes.

The Sagnac effect is re-examined using Finslerian geometry.

The paper studies solutions to a nonlinear equation on Finsler manifolds with gradient estimates and Harnack inequalities.

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integr…

A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that …

For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or bilipshitz deformation …

The paper explores Finsler-type objects and their variational problems on spacetimes.

The Finslerian extension of the Euclidean metric is proposed and studied under rigorous conditions that the associated indicatrix is regular and convex. The relativistic pseudo-Euclidean metric is extended, too. The extensions show distinct violation of the $T-$parity, so that the future-past asymmetry of the physical …

Global gradient estimates for Fisher-KPP equation on Finsler metric measure spaces.

Unified geometry for relativity and beyond.

The paper proposes a notion of volume element for Finsler spaces with metrics of Lorentzian signature, equipped with a time orientation. This notion is based on a slight modification of the idea of Holmes-Thompson volume element working for positive definite Finsler metrics and can be used in field-theoretical applicat…

Established a Hardy inequality on Finsler manifolds.

Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary $\partial V$ for a standard conformally stationary spacetime V = R x M, suggests a natural compactification $M_B$ associated to any Riemannian metric on M or, more generally, to any Fin…

The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are given and some recent results are presented. They are a generalization of Einste…

Biharmonic curves are a generalization of geodesics, with applications in elasticity theory and various branches of computer science. The paper proposes a first study of biharmonic curves in spaces with Finslerian geometry, covering the following topics: a deduction of their equations, existence of non-geodesic biharmo…

Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of co…

Introduces a natural parallel translation for navigation data.

The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces. Quasi-Einstein metrics serve also as solution to the Ricci flow equation. Here, the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is …