arXiv research

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

Study traveling waves in hyperbolic space for Fisher-KPP equations.

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

Researchers prove constant solutions for a specific Finslerian equation.

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

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

Paper develops methods for estimating gradients of Finslerian Schrödinger equations.

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…

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…

Study on nonholonomic mechanics and sub-Finsler geometry.

Finslerian graph neural networks recover nonlinear diffusion geometry

Introduces Finslerian convolution metrics and their properties.

Develops Palatini formalism for pseudo-Finsler metrics, recovering classical results.

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 …

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

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

Lecture notes on Finslerian geometry.

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…

Develops geometric framework for dissipative field equations.

Classification of Finslerian spaces with nontrivial concircular transformations.

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

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.

We study the variational problem for $N$-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler-Lagrange equations for this type of variational problem…

Recently, the behavior of different epidemic models and their relation both to different types of geometries and to some biological models has been revisited . Path equations representing the behavior of epidemic models and their corresponding deviation vectors are examined. A comparison between paths and their deviati…

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by clas…

The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus appl…

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

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

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…

Anisotropy of a space naturally leads to direction dependent electromagnetic tensors and electromagnetic potentials. Starting from this idea and using variational approaches and exterior derivative formalism, we extend some of the classical equations of electromagnetism to anisotropic (Finslerian) spaces. The results d…

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

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

Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.

The class of the two-axes pseudo-Finslerian metrics which is specified by the condition of the angle-separation in the involved characteristic functions is proposed and studied. The complete Total Set of algebraic and differential equations is derived in all rigor which are necessary and sufficient in order that a pseu…

The notions of bienergy of a smooth mapping and of biharmonic map between Riemannian manifolds are extended to the case when the domain is Finslerian. We determine the first and the second variation of the bienergy functional, the equations of Finsler-to-Riemann biharmonic maps and some specific examples. Two notable r…

Unified geometry for relativity and beyond.

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…

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 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 …

Study on generalized $m$-Kropina metrics in modified gravity and cosmology.

The Sagnac effect is re-examined using Finslerian geometry.

Our previous exploration of the $\cE_g^{PD}$-geometry has shown that the field is promising. Namely, the $\cE_g^{PD}$-approach is amenable to development of novel trends in relativistic and metric differential geometry and can particularly be effective in context of the Finslerian or Minkowskian Geometries. The main po…

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 $Ω$…

New findings on lightconvex boundaries in Finslerian spacetimes.

Locally classifies 4D spherical symmetric Finsler spaces.

In this work we study Berwald spacetimes and their vacuum dynamics, where the latter are based on a Finsler generalization of the Einstein's equations derived from an action on the unit tangent bundle. In particular, we consider a specific class of spacetimes which are non-flat generalizations of the very special relat…

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduc…

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